Difference between revisions of "MAT4233"

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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]
 
*[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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https://en.wikipedia.org/wiki/Coproduct
 
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*Subgroups of cyclic groups and their generators
 
*Subgroups of cyclic groups and their generators
 
*Subgroup lattice
 
*Subgroup lattice
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*Chinese Remainder Theorem
 
*Chinese Remainder Theorem
 
*Internal products
 
*Internal products
*Coproducts
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*[https://en.wikipedia.org/wiki/Coproduct Coproducts]
 
*Universal property of coproduct
 
*Universal property of coproduct
 
*Classification of finitely generated Abelian groups (without proof)
 
*Classification of finitely generated Abelian groups (without proof)

Revision as of 17:47, 12 July 2020

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, Contemporary abstract algebra (8e) Houghton Mifflin

Topics List

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
  • change of basis S in R^n gives an inner automorphism of GL(n,R): X -> S^(-1).X.S
  • C -> C: z -> 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 complex numbers

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/nZ
  • 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
  • ZxZ
  • 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
  • mZ < Z
  • <2, x> = 2Z[x]+xZ[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
  • 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