Department of Mathematics at UTSA - User contributions [en]

MAT4233

2020-07-13T00:44:53Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||'''0:''' 2, 4, 10, 14, 15, 17, 18 (show work), 20, 21, 22, 27, 28.

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms*'''14:''' 4, 6, 10.
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||'''2:''' 2, 3, 19, 22, 23, 25, 26, 28, 35, 37, 39, 47.

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||
*'''10:''' 1, 2, 3, 6, 15, 21, 24, 25.
*'''6:''' 3, 8, 14, 17, 35, 39, 49, 58, 61.

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||'''3:''' 4, 7, 11, 28, 29, 32.

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||'''4:''' 10, 14, 18, 32, 52.

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5, 1
||
||
||
||
*'''5:''' 3, 5, 11, 21, 22, 24, 26, 32, 36, 38.
*'''1:''' 1, 2, 13, 22.

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||
*'''7:''' 17, 20, 24, 26, 33, 43, 60.
*'''9:''' 12, 13, 14.

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*free product of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||
*'''8:''' 2, 5, 12, 54, 55, 56.
*'''9:''' 22, 24, 28, 31, 32, 48, 49, 50.
*'''11:''' 3, 4.

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*, '''Z'''_n* = U(n)
*polynomial rings
||
*'''12:''' 4, 6, 7, 12, 13, 22, 26.
*'''13:''' 7, 12.
*'''15:''' 20, 8, 12, 13, 19.

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||'''14:''' 4, 6, 8, 10, 28, 31.

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||
*'''16:''' 19, 20, 26, 32, 34.
*'''17:''' 2, 4, 6, 11, 13, 14, 15, 16.
|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-13T00:38:24Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||'''0:''' 2, 4, 10, 14, 15, 17, 18 (show work), 20, 21, 22, 27, 28.

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||'''2:''' 2, 3, 19, 22, 23, 25, 26, 28, 35, 37, 39, 47.

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||
*'''10:''' 1, 2, 3, 6, 15, 21, 25.
*'''6:''' 3, 8, 14, 17, 35, 39, 49, 58, 61.

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||'''3:''' 4, 7, 11, 28, 29, 32.

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||'''4:''' 10, 14, 18, 32, 52.

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5, 1
||
||
||
||
*'''5:''' 3, 5, 11, 21, 22, 24, 26, 32, 36, 38.
*'''1:''' 1, 2, 13, 22.

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||
*'''7:''' 17, 20, 24, 26, 33, 43, 60.
*'''9:''' 12, 13, 14.

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*free product of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||
*'''8:''' 2, 5, 12, 54, 55, 56.
*'''9:''' 22, 24, 28, 31, 32, 48, 49, 50.
*'''11:''' 3, 4.

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*, '''Z'''_n* = U(n)
*polynomial rings
||
*'''12:''' 4, 6, 7, 22, 26.
*'''13:''' 7.

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||'''14:''' 4, 6, 10.

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-13T00:36:01Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||'''0:''' 2, 4, 10, 14, 15, 17, 18 (show work), 20, 21, 22, 27, 28.

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||'''2:''' 2, 3, 19, 22, 23, 25, 26, 28, 35, 37, 39, 47.

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||
*'''10:''' 1, 2, 3, 6, 15, 21, 25.
*'''6:''' 3, 8, 14, 17, 35, 39, 49, 58, 61.

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||'''3:''' 4, 7, 11, 28, 29, 32.

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||'''4:''' 10, 14, 18, 32, 52.

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5, 1
||
||
||
||
*'''5:''' 3, 5, 11, 21, 22, 24, 26, 32, 36, 38.
*'''1:''' 1, 2, 13, 22.

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||
*'''7:''' 17, 20, 24, 26, 33, 43, 60.
*'''9:''' 12, 13, 14.

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*free product of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||
*'''8:''' 2, 5, 12, 54, 55, 56.
*'''9:''' 22, 24, 28, 31, 32, 48, 49, 50.
*'''11:''' 3, 4.

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*, '''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-13T00:31:18Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||'''0:''' 2, 4, 10, 14, 15, 17, 18 (show work), 20, 21, 22, 27, 28.

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||'''2:''' 2, 3, 19, 22, 23, 25, 26, 28, 35, 37, 39, 47.

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||
*'''10:''' 1, 2, 3, 6, 15, 21, 25.
*'''6:''' 3, 8, 14, 17, 35, 39, 49, 58, 61.
|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||'''3:''' 4, 7, 11, 28, 29, 32.

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||'''4:''' 10, 14, 18, 32, 52.

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5, 1
||
||
||
||
*'''5:''' 3, 5, 11, 21, 22, 24, 26, 32, 36, 38.
*'''1:''' 1, 2, 13, 22.
|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*free product of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*, '''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-13T00:28:27Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||0: 2, 4, 10, 14, 15, 17, 18 (show work), 20, 21, 22, 27, 28.

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||2: 2, 3, 19, 22, 23, 25, 26, 28, 35, 37, 39, 47.

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||
*10: 1, 2, 3, 6, 15, 21, 25.
*6: 3, 8, 14, 17, 35, 39, 49, 58, 61.
|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||3: 4, 7, 11, 28, 29, 32.

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||4: 10, 14, 18, 32, 52.

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5, 1
||
||
||
||1: 1, 2, 13, 22.

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*free product of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*, '''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-13T00:17:15Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||0: 2, 4, 10, 14, 15, 17, 18 (show work), 20, 21, 22, 27, 28.

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||2: 2, 3, 19, 22, 23, 25, 26, 28, 35, 37, 39, 47.

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5, 1
||
||
||
||1: 1, 2, 13, 22.

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*free product of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*, '''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-13T00:15:25Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||2: 2, 3, 19, 22, 23, 25, 26, 28, 35, 37, 39, 47.

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5, 1
||
||
||
||1: 1, 2, 13, 22.

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*free product of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*, '''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-13T00:13:44Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the d|-
|7
||
*Catch up and review
*Midterm 1
efinition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||2: 2, 3, 19, 22, 23, 25, 26, 28, 35, 37, 39, 47.

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5, 1
||
||
||
||1: 1, 2, 13, 22.

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*free product of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*, '''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-12T23:57:57Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the d|-
|7
||
*Catch up and review
*Midterm 1
efinition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5
||
||
||
||

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*free product of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*, '''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-12T23:56:14Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the d|-
|7
||
*Catch up and review
*Midterm 1
efinition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5
||
||
||
||

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*free product of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*
*'''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-12T23:54:32Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the d|-
|7
||
*Catch up and review
*Midterm 1
efinition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5
||
||
||
||

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*coproduct of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*
*'''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-12T23:53:11Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||

|-
|7
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5
||
||
||
||

|-
|8
||
*Catch up and review
*Midterm 1

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products
||
*External direct product
*Universal property of direct product
*Chinese Remainder Theorem
*Internal direct product
*[https://en.wikipedia.org/wiki/Coproduct Free product]
*Universal property of free product
*Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*coproduct of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*
*'''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-12T23:47:15Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||
https://en.wikipedia.org/wiki/Coproduct
|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4
||
||
||
||

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5
||
||
||
||

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products and coproducts
||
*External products
*Universal property of product
*Chinese Remainder Theorem
*Internal products
*[https://en.wikipedia.org/wiki/Coproduct Coproducts]
*Universal property of coproduct
*Classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*coproduct of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*
*'''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-12T23:45:05Z

Dmitry.gokhman: /* Topics List */

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sprs.pdf Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4Bézout
||
||
||
||

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5
||
||
||
||

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products and coproducts
||
*External products
*Universal property of product
*Chinese Remainder Theorem
*Internal products
*Coproducts
*Universal property of coproduct
*Classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*coproduct of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*
*'''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-12T23:40:54Z

Dmitry.gokhman:

Modern Abstract Algebra (3-0) 3 Credit Hours

==Description==

The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.

==Evaluation==

* Two midterms (for classes that meet twice a week) and an optional final.

* Exam score is the best of final score and midterm average.

* Students will have access to several past exams for practice.

==Text==
J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises

|-
|1
||'''Z'''
||
*The natural order on '''N''' and the well ordering principle
*Mathematical induction
*Construction of '''Z''' and its properties (graph the equivalence classes)
*Division algorithm
*Congruence mod m
*Algebra on the quotient set '''Z'''_m
*GCD, LCM, Bézout
*Primes, Euclid's Lemma
*Fundamental Theorem of Arithmetic
||0
||
*Sets
*Partitions
*Equivalence relations and classes
*Functions
*Images and preimages
||
*Review of known facts about '''Z'''
*A concrete introduction to techniques of abstract algebra
||
*Equivalence classes are partitions
*If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
*Introduce congruence using the remainder function.
*Congruence classes mod 3
*Extended Euclid's algorithm
||

|-
|2
|| Groups
||
*Symmetries
*Properties of composition
*Definition of a group
*Elementary proofs with groups:
**uniqueness of identity
**uniqueness of inverses
**cancellation
**shortcuts to establishing group axioms
*Foundational examples with Cayley tables
||2
||Sets and functions
||
*Motivation for the concept of a group
*Learn the definition of a group
*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
*Starting to build a catalog of examples of groups
*Learn to construct and read Cayley tables
||
*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
*'''R'''^n, M(n,'''R''')
*symmetric group S_n
*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
*correspondence of '''Z'''_2 with {-1, 1} and with S_2
*functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
*free group on a finite set
||

|-
|3
||Homomorphisms
||
*Cayley's theorem
*Homomorphisms of groups
*Isomorphisms and their inverses
*Automorphisms
*Examples
||10, 6
||
*Functions
*Groups
*Matrix multiplication
*Change of basis for matrices
||
*General framework for thinking of groups as symmetries and motivation for homomorphisms
*Learn the definitions of homomorphism and isomorphism
*Prove that homomorphisms preserve powers.
*Starting to build a catalog of examples of homomorphisms.
||
*'''R''' -> '''R''': x -> ax
*'''R'''^n -> '''R'''^n: v -> Av
*M(n,'''R''') -> M(n,'''R'''): X -> AX
*'''R'''* -> '''R'''*: x -> x^n
*'''R''' -> '''R'''+: x -> a^x (a>0)
*determinant: GL(n,'''R''') -> '''R'''*
*inclusions
*natural projection '''Z''' -> '''Z'''_2
*evaluation {X -> G} -> G: f -> f(a)
*'''Z''' -> '''Z''': k -> -k
*Aut('''Z'''_2) is trivial
*Aut('''Z'''_3) is isomorphic to '''Z'''_2
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/linear_maps_in_coordinates.pdf change of basis] S in '''R'''^n gives an inner automorphism of GL(n,'''R'''): X -> S^(-1).X.S
*'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
||

|-
|4
||Subgroups
||
*Definition of a subgroup
*Subgroup tests
*Automatic closure under inverses for finite subgroups
*Subgroups generated by a subset
*Examples
*Images and preimages under a homomorphism are subgroups.
*Fibers as cosets of the kernel
*First Isomorphism Theorem
*Examples
||3, 10
||
*Groups
*Functions
*Equivalence relations and classes
||
*Learn how to identify subgroups, with proofs.
*Learn how to obtain new groups from old via homomorphisms.
*Learn how to prove a homomorphism is one-to-one by using the kernel.
||
*Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
||

|-
|5
||Groups in Linear Algebra and Complex Variable
||
* Euclidean space as an additive group
* Null space and column space of a linear map
*Solutions to linear inhomogeneous systems
* Invertible linear transformations and matrices, GL(n,'''R''')
* Determinant: homomorphism, similarity invariance, geometrical interpretation.
* Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
||
||
||
||
*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
*'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
*[https://en.wikipedia.org/wiki/Linear_fractional_transformation Möbius transformations] on the [https://en.wikipedia.org/wiki/Riemann_sphere Riemann sphere]
||

|-
|6
||Cyclic groups
||
*Order of a group, order of an element
*Defining homomorphisms on '''Z''' (free group)
*Classification of cyclic groups
*Subgroups of cyclic groups and their generators
*Subgroup lattice
||4Bézout
||
||
||
||

|-
|7
||
*Catch up and review
*Midterm 1

|-
|8
||Permutations
||
*Cycle notation
*D_n as a subgroup of S_n
*Factoring into disjoint cycles
*Ruffini's theorem
*Cyclic subgroups, powers of a permutation
*Parity, A_n < S_n
||5
||
||
||
||

|-
|9
||Cosets
||
*Cosets as equivalence classes
*Lagrange's theorem
*Fermat's little theorem
*Euler's theorem
*Normal subgroups
*Factor groups
*Universal property of factor groups
*First Isomorphism theorem revisited
||7, 9
||
||
||
*cosets of <(1,2)> in S_3
*cosets of a flip in D_4
*inverse images of subgroups are normal, kernels
*A_n is normal in S_n
*rotations in D_n
*'''Z'''/n'''Z'''
*'''R'''/'''Z'''
||

|-
|10
||Products and coproducts
||
*External products
*Universal property of product
*Chinese Remainder Theorem
*Internal products
*Coproducts
*Universal property of coproduct
*Classification of finitely generated Abelian groups (without proof)
||8, 9, 11
||
||
||
*'''Z'''x'''Z'''
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
*coproduct of '''Z''' with itself in groups and in Abelian groups
*free group on a set
*free Abelian group on a set
||

|-
|11
||Rings
||
*Motivation and definition
*Properties
*Subrings
*Integral domains
*Fields
*Characteristic
*Ring homomorphisms
*Examples
||12, 13, 15, 16
||
||
||
*'''Z''' and other number systems
*R*
*'''Z'''_n* = U(n)
*polynomial rings
||

|-
|12
||Ideals and factor rings
||
*Ideals
*Ideals generated by a set, principal ideals
*Images and preimages of ideals are ideals
*Factor rings
*Prime ideals
*Maximal ideals
*Localization, field of quotients
||14
||
||
||
*m'''Z''' < '''Z'''
*<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
*Hausdorff Maximality Principle
*

*'''Q'''[x]/<x^2-2>
*'''R'''[x]/<x^2+1>
*'''Z''' -> '''Q'''
*polynomials -> rational functions
||

|-
|13
||Factorization
||
*Division algorithm for F[x]
*F[x] is a PID
*Factorization of polynomials
*[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
*Tests, Eisenstein's criterion
*Irreducibles and associates
*'''Z'''[x] is a UFD
||16, 17, 18
||
||
||
*In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
||

|-
|14
||
*Catch up and review
*Midterm 2
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT4233

2020-07-11T21:59:55Z

Dmitry.gokhman:

MAT4233

2020-07-11T21:58:55Z

Dmitry.gokhman:

MAT4233

2020-07-11T20:21:07Z

Dmitry.gokhman: Created page with "Modern Abstract Algebra (3-0) 3 Credit Hours ==Description== The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction o..."

MAT3013

2020-07-10T23:59:19Z

Dmitry.gokhman:

Foundations of Mathematics (3-0) 3 Credit Hours

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite skills !! Learning outcomes !! Examples

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

MAT3013

2020-07-10T23:50:59Z

Dmitry.gokhman:

Foundations of Mathematics (3-0) 3 Credit Hours

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite Skills !! Learning Outcomes !! Examples !

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days
|}

==See also==

* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]

Main Page

2020-07-10T23:41:52Z

Dmitry.gokhman:

<strong>UTSA Department of Mathematics</strong>

* [[MAT1023]] College Algebra
* [[MAT1043]] Introduction to Mathematics
* [[MAT1053]] Algebra for Business
* [[MAT1073]] College Algebra for Scientists and Engineers
* [[MAT1093]] Precalculus
* [[MAT1133]] Calculus for Business
* [[MAT1193]] Calculus for Biosciences
* [[MAT1214]] Calculus I
* [[MAT1224]] Calculus II
* [[MAT2214]] Calculus III
* [[MAT2233]] Linear Algebra
* [[MAT3013]] Foundations of Mathematics
* [[MAT3613]] Differential Equations I
* [[MAT4233]] Modern Abstract Algebra

Main Page

2020-07-10T23:39:55Z

Dmitry.gokhman:

<strong>UTSA Department of Mathematics</strong>

* [[MAT1023]] College Algebra
* [[MAT1043]]
* [[MAT1053]] Algebra for Business
* [[MAT1073]] College Algebra for Scientists and Engineers
* [[MAT1093]] Precalculus
* [[MAT1133]]
* [[MAT1193]] Calculus for Biosciences
* [[MAT1214]] Calculus I
* [[MAT1224]] Calculus II
* [[MAT2214]] Calculus III
* [[MAT2233]] Linear Algebra
* [[MAT3013]] Foundations of Mathematics
* [[MAT3613]] Differential Equations I
* [[MAT4233]] Modern Abstract Algebra

MAT3013

2020-07-10T23:16:25Z

Dmitry.gokhman: /* Topics List */

Foundations of Mathematics (3-0) 3 Credit Hours

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite Skills !! Learning Outcomes !! Examples !

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
||
|-
|15
||
*Catch up and review for final
* Study days

MAT3013

2020-07-10T23:15:40Z

Dmitry.gokhman:

Foundations of Mathematics (3-0) 3 Credit Hours

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite Skills !! Learning Outcomes !! Examples !

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
|-
|15
||
*Catch up and review for final
* Study days

MAT3013

2020-07-10T23:15:21Z

Dmitry.gokhman: Created page with "MAT3013 Foundations of Mathematics (3-0) 3 Credit Hours ==Description== Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major s..."

MAT3013 Foundations of Mathematics (3-0) 3 Credit Hours

==Description==

Foundations of Mathematics is a pivotal course for mathematics majors. It serves as the first major step towards modern mathematics
of rigorous proofs and a true pre-requisite to real analysis and abstract algebra. Up to this point students are asked to do few proofs
(notably geometry and perhaps some epsilon-delta in calculus). The course particularly emphasizes set-theoretical constructions, such
as functions, composition, inversion, forward and inverse images, relations, equivalence relations, partial orders, quotient sets and
products and unions of sets, vital to further work in mathematics.

==Evaluation==

* No makeup exams are offered.

* An absence may be excused if sufficient evidence of extenuating circumstances is provided. In this case, the final exam grade
could be used as the grade for the missed exam.

* Students will have access to several past exams for practice.

==Text==

D. Smith, M. Eggen, R. St. Andre, ''A Transition to Advanced Mathematics'' (7e), Brooks/Cole

==Topics List==
{| class="wikitable sortable"
! Week !! Session !! Topics !! Section !! Prerequisite Skills !! Learning Outcomes !! Examples !

|-
|1

|| Introduction

||
* Historical remarks
* Overview of the course and its goals
* Ideas of proofs and logic
* Axioms and propositions

|| 1.1

||

||

* Motivation for rigorous
mathematics from a
historical perspective
* An understanding of where
and why this course is
going
|-
|2
|| Propositional logic
||
* Logical operators
* Truth values
* Truth tables
* Quantifiers

|| 1.2-3
||
|| Gain the prerequisites for
writing and evaluating
proofs.
||
* connectives
* conditionals
* biconditionals

|-
|3
|| Proof methods
|| Methods for proofs
|| 1.4-6
|| Propositional logic
|| Start proving elementary results.
||
* direct proofs
* ''modus ponens''
* proofs by contradiction

|-
|4
|| Set theory
||
* Basic concepts
* Operations and constructions with sets
|| 2.1-3
|| Basic concepts of set theory
|| How to start working with sets
||
* notation
* subsets
* proving sets are equal
* unions, intersections, complements

|-
|5
|| Induction and counting
||
* Mathematical induction
* Counting principles
||2.4-6
||Natural numbers
||
* Learn constructive proofs and reasoning.
* Learn basic counting principles of discrete mathematics.
||
* sums of consecutive powers
* other induction proofs
* well ordering principle
* inclusion-exclusion principle
|-
|6
||
* Catch up and review
* Midterm 1

|-
|7
||Relations 1
||
* Cartesian products and their subsets
* Equivalence relations
||3.1-3
||Set theory
||Gain basic concepts about relations.
||
* modular congruence
* gluing sets
|-
|8
||Relations 2
||
* Partial orders
* Graphs
||3.4-5
||Relations 1
||
* Familiarize with ordering.
* Learn how to use graph representations of relations.
|| partial ordering of the power set under inclusion
|-
|9
||Functions 1
||
* Functions
* Constructions with functions
||4.1-2
||
* Relations
* Function sense (precalculus)
|| Gain basic rigorous knowledge of functions.
||
functional composition
|-
|10
||Functions 2
||
* One-to-one
* Onto
* Compositional inverse
||4.3-4
||Functions 1
||
* Determine whether a function is one-to-one of onto, with proofs.
* Finding inverses
||
* examples with finite sets
* many precalculus examples

|-
|11
||Functions 3
||
* Images of subsets
* Preimages of subsets
* Sequences
||4.5-6
||Functions 2
||Find images and preimages of subsets under functions, with proofs.
||
* examples with finite sets
* many precalculus examples

|-
|12
||
* Catch up and review
* Midterm 2
|-
|13
||Cardinality 1
||
* Finite and infinite sets
* Equivalent sets
||5.1-2
||Sets and functions
||
* Learn classification of sets by size.
* Generalizing the concept of size to infinite sets
||

|-
|14
||Cardinality 2
|| Countable and uncountable sets
||5.3-5
||Cardinality 1
||Learn properties of countable sets.
|-
|15
||
*Catch up and review for final
* Study days

Main Page

2020-07-10T16:35:46Z

Dmitry.gokhman: