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| − | Modern Abstract Algebra (3-0) 3 Credit Hours | + | ==Modern Abstract Algebra== |
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| − | ==Description==
| + | [https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 4233. Modern Abstract Algebra]. (3-0) 3 Credit Hours. |
| | | | |
| − | 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.
| + | Prerequisites: [[MAT3233]]. |
| − | The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.
| + | Basic properties and examples of semigroups, monoids, and groups, detailed study of permutation, dihedral, and congruence groups, cyclic groups, normal subgroups, quotient groups, homomorphism, isomorphism theorems, direct products of groups, rings and fields and their basic properties, ideals, polynomial rings. Generally offered: Spring. Differential Tuition: $150. |
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| − | ==Evaluation==
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| − |
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| − | * Two midterms (for classes that meet twice a week) and an optional final.
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| − |
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| − | * Exam score is the best of final score and midterm average.
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| − |
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| − | * Students will have access to several past exams for practice.
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| | | | |
| | ==Text== | | ==Text== |
| − | J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin
| + | Charles C. Pinter. [https://store.doverpublications.com/0486474178.html ''A Book of Abstract Algebra.''] (Reprint of 2nd. ed. originally published in 1990) Dover Publications, NY (2010). ISBN-10: 0-486-47417-8. |
| | | | |
| − | ==Topics List== | + | ==Topics== |
| | {| class="wikitable sortable" | | {| class="wikitable sortable" |
| − | ! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises | + | ! Date !! Sections !! Topics !! Student Learning Outcomes |
| | + | |- |
| | + | | |
| | + | 1 |
| | + | || |
| | + | 1 & 2 |
| | + | || |
| | + | Binary operations on sets. |
| | + | || |
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| | |- | | |- |
| − | |1 | + | | |
| − | ||'''Z'''
| + | 2 |
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| − | *The natural order on '''N''' and the well ordering principle
| + | 3 & 4 |
| − | *Mathematical induction
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| − | *Construction of '''Z''' and its properties (graph the equivalence classes)
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| − | *Division algorithm
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| − | *Congruence mod m
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| − | *Algebra on the quotient set '''Z'''_m
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| − | *GCD, LCM, Bézout
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| − | *Primes, Euclid's Lemma
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| − | *Fundamental Theorem of Arithmetic
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| − | ||0
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| − | *Sets
| + | Groups: Definition and elementary properties. |
| − | *Partitions
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| − | *Equivalence relations and classes
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| − | *Functions
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| − | *Images and preimages
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| − | ||
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| − | *Review of known facts about '''Z'''
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| − | *A concrete introduction to techniques of abstract algebra
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| − | ||
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| − | *Equivalence classes are partitions
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| − | *If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
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| − | *Introduce congruence using the remainder function.
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| − | *Congruence classes mod 3
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| − | *Extended Euclid's algorithm
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| | |- | | |- |
| − | |2 | + | | |
| − | || Groups
| + | 3 |
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| − | *Symmetries
| + | 5 |
| − | *Properties of composition
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| − | *Definition of a group
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| − | *Elementary proofs with groups:
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| − | **uniqueness of identity
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| − | **uniqueness of inverses
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| − | **cancellation
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| − | **shortcuts to establishing group axioms
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| − | *Foundational examples with Cayley tables
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| − | ||2
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| − | ||Sets and functions
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| | || | | || |
| − | *Motivation for the concept of a group
| + | Subgroups. |
| − | *Learn the d|-
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| − | |7
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| − | ||
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| − | *Catch up and review
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| − | *Midterm 1
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| − | efinition of a group
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| − | *Learn basic automatic properties of groups (with proofs) for later use as shortcuts
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| − | *Starting to build a catalog of examples of groups
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| − | *Learn to construct and read Cayley tables
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| − | ||
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| − | *'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
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| − | *'''R'''^n, M(n,'''R''')
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| − | *symmetric group S_n
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| − | *'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
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| − | *correspondence of '''Z'''_2 with {-1, 1} and with S_2
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| − | *functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
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| − | *free group on a finite set
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| | |- | | |- |
| − | |3 | + | | |
| − | ||Homomorphisms
| + | 4 |
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| − | *Cayley's theorem
| + | 7 & 8 |
| − | *Homomorphisms of groups
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| − | *Isomorphisms and their inverses
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| − | *Automorphisms
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| − | *Examples
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| − | ||10, 6
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| − | *Functions
| + | Groups of Permutations. Permutations of finite sets. |
| − | *Groups
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| − | *Matrix multiplication
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| − | *Change of basis for matrices
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| − | ||
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| − | *General framework for thinking of groups as symmetries and motivation for homomorphisms
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| − | *Learn the definitions of homomorphism and isomorphism
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| − | *Prove that homomorphisms preserve powers.
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| − | *Starting to build a catalog of examples of homomorphisms.
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| − | ||
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| − | *'''R''' -> '''R''': x -> ax
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| − | *'''R'''^n -> '''R'''^n: v -> Av
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| − | *M(n,'''R''') -> M(n,'''R'''): X -> AX
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| − | *'''R'''* -> '''R'''*: x -> x^n
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| − | *'''R''' -> '''R'''+: x -> a^x (a>0)
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| − | *determinant: GL(n,'''R''') -> '''R'''*
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| − | *inclusions
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| − | *natural projection '''Z''' -> '''Z'''_2
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| − | *evaluation {X -> G} -> G: f -> f(a)
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| − | *'''Z''' -> '''Z''': k -> -k
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| − | *Aut('''Z'''_2) is trivial
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| − | *Aut('''Z'''_3) is isomorphic to '''Z'''_2
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| − | *[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
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| − | *'''C''' -> '''C''': z -> [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex conjugate] of z
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| − | |4 | + | | |
| − | ||Subgroups
| + | 5 |
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| − | *Definition of a subgroup
| + | 9 & 10 |
| − | *Subgroup tests
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| − | *Automatic closure under inverses for finite subgroups
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| − | *Subgroups generated by a subset
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| − | *Examples
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| − | *Images and preimages under a homomorphism are subgroups.
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| − | *Fibers as cosets of the kernel
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| − | *First Isomorphism Theorem
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| − | *Examples
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| − | ||3, 10
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| − | *Groups
| + | Group isomorphisms and orders of elements. |
| − | *Functions
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| − | *Equivalence relations and classes
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| − | ||
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| − | *Learn how to identify subgroups, with proofs.
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| − | *Learn how to obtain new groups from old via homomorphisms.
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| − | *Learn how to prove a homomorphism is one-to-one by using the kernel.
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| − | ||
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| − | *Cyclic subgroups <x>={x^k: k in '''Z'''} or xZ={xk: k in '''Z'''}
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| | |- | | |- |
| − | |5 | + | | |
| − | ||Groups in Linear Algebra and Complex Variable
| + | 6 |
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| − | * Euclidean space as an additive group
| + | --- |
| − | * Null space and column space of a linear map
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| − | *Solutions to linear inhomogeneous systems
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| − | * Invertible linear transformations and matrices, GL(n,'''R''')
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| − | * Determinant: homomorphism, similarity invariance, geometrical interpretation.
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| − | * Additive and multiplicative subgroups of [http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf complex numbers]
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| | + | Review. First midterm exam. |
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| | + | |
| | + | |- |
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| | + | 7 |
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| | + | 11 |
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| − | *GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
| + | Cyclic groups. |
| − | *'''C''', '''C'''*, S^1, ''n''-th roots of unity, D_n as a subgroup of O(2,'''R'''^2)
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| − | *[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]
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| | |- | | |- |
| − | |6 | + | | |
| − | ||Cyclic groups
| + | 8 |
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| − | *Order of a group, order of an element
| + | 13 |
| − | *Defining homomorphisms on '''Z''' (free group)
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| − | *Classification of cyclic groups
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| − | *Subgroups of cyclic groups and their generators
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| − | *Subgroup lattice
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| − | ||4
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| | + | Counting cosets. |
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| | + | |
| | + | |- |
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| | + | 9 |
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| | + | 14 & 15 |
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| − | | + | Homomorphisms and quotient groups. |
| − | |-
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| − | |7
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| − | *Catch up and review
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| − | *Midterm 1
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| | | | |
| | |- | | |- |
| − | |8 | + | | |
| − | ||Permutations
| + | 10 |
| − | ||
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| − | *Cycle notation
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| − | *D_n as a subgroup of S_n
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| − | *Factoring into disjoint cycles
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| − | *Ruffini's theorem
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| − | *Cyclic subgroups, powers of a permutation
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| − | *Parity, A_n < S_n
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| − | ||5
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| − | ||
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| | || | | || |
| | + | --- |
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| | + | Review. Second midterm exam. |
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| | |- | | |- |
| − | |9 | + | | |
| − | ||Cosets
| + | 11 |
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| − | *Cosets as equivalence classes
| + | 16 |
| − | *Lagrange's theorem
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| − | *Fermat's little theorem
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| − | *Euler's theorem
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| − | *Normal subgroups
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| − | *Factor groups
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| − | *Universal property of factor groups
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| − | *First Isomorphism theorem revisited
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| − | ||7, 9
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| − | ||
| + | The fundamental theorem of homomorphism. |
| − | ||
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| − | *cosets of <(1,2)> in S_3
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| − | *cosets of a flip in D_4
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| − | *inverse images of subgroups are normal, kernels
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| − | *A_n is normal in S_n
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| − | *rotations in D_n
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| − | *'''Z'''/n'''Z'''
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| − | *'''R'''/'''Z'''
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| | |- | | |- |
| − | |10 | + | | |
| − | ||Products
| + | 12 |
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| − | *External direct product
| + | 17 |
| − | *Universal property of direct product
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| − | *Chinese Remainder Theorem
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| − | *Internal direct product
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| − | *[https://en.wikipedia.org/wiki/Coproduct Free product]
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| − | *Universal property of free product
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| − | *Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
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| − | ||8, 9, 11
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| − | ||
| + | Rings and ideals. |
| − | ||
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| − | *'''Z'''x'''Z'''
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| − | *[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/sun.pdf public key cryptography]
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| − | *free product of '''Z''' with itself in groups and in Abelian groups
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| − | *free group on a set
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| − | *free Abelian group on a set
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| | |- | | |- |
| − | |11 | + | | |
| − | ||Rings
| + | 13 |
| − | ||
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| − | *Motivation and definition
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| − | *Properties
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| − | *Subrings
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| − | *Integral domains
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| − | *Fields
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| − | *Characteristic
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| − | *Ring homomorphisms
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| − | *Examples
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| − | ||12, 13, 15, 16
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| − | ||
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| | + | 18 |
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| − | *'''Z''' and other number systems
| + | Ring homomorphisms. |
| − | *R*, '''Z'''_n* = U(n)
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| − | *polynomial rings
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| | |- | | |- |
| − | |12 | + | | |
| − | ||Ideals and factor rings
| + | 14 |
| − | ||
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| − | *Ideals
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| − | *Ideals generated by a set, principal ideals
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| − | *Images and preimages of ideals are ideals
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| − | *Factor rings
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| − | *Prime ideals
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| − | *Maximal ideals
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| − | *Localization, field of quotients
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| − | ||14
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| | + | 19 & 20 |
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| | + | Quotient rings and integral domains. |
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| − | *m'''Z''' < '''Z'''
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| − | *<2, x> = 2'''Z'''[x]+x'''Z'''[x] < '''Z'''[x]
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| − | *Hausdorff Maximality Principle
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| − | *
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| | | | |
| − | *'''Q'''[x]/<x^2-2>
| + | <!-- |- --> |
| − | *'''R'''[x]/<x^2+1>
| + | <!-- | --> |
| − | *'''Z''' -> '''Q'''
| + | <!-- 15 --> |
| − | *polynomials -> rational functions
| + | <!-- || --> |
| − | || | + | <!-- 21 & 22 --> |
| | + | <!-- || --> |
| | + | <!-- The integers. Factorization into primes. --> |
| | + | <!-- || --> |
| | + | <!-- |- --> |
| | + | <!-- | --> |
| | + | <!-- 14 --> |
| | + | <!-- || --> |
| | + | <!-- 23 --> |
| | + | <!-- || --> |
| | + | <!-- Elements of number theory. --> |
| | + | <!-- || --> |
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| | |- | | |- |
| − | |13 | + | | |
| − | ||Factorization
| + | 15 |
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| − | *Division algorithm for F[x]
| + | --- |
| − | *F[x] is a PID
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| − | *Factorization of polynomials
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| − | *[http://zeta.math.utsa.edu/~gokhman/ftp//courses/notes/fta.pdf Fundamental Theorem of Algebra ]
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| − | *Tests, Eisenstein's criterion
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| − | *Irreducibles and associates
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| − | *'''Z'''[x] is a UFD
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| − | ||16, 17, 18
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| | || | | || |
| − | ||
| + | Wrap-up and review. Student Study Day. |
| − | ||
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| − | *In '''Z'''[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
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| | |- | | |- |
| − | |14
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| − | ||
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| − | *Catch up and review
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| − | *Midterm 2
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| − | |-
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| − | |15
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| − | ||
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| − | *Catch up and review for final
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| − | * Study days
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| | |} | | |} |
| − |
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| − | ==See also==
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| − |
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| − | * [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]
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