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
- Midterms 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
|
Introduction to 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
|
|
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
- Planting seeds for later concepts of subgroup, factor group, homomorphism, isomorphism, ring
|
- 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
|
|
| 2
|
Introduction to homomorphisms
|
- Homomorphisms of groups
- Isomorphisms and their inverses
- Examples
- Cayley's theorem
|
|
- Functions
- Groups
- Matrix multiplication
|
- Learn the definitions of homomorphism and isomorphism
- Starting to build a catalog of examples of homomorphisms
- Prove that homomorphisms preserve powers for later use.
- General framework for thinking of groups as symmetries
|
- 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
- inclusions
- natural projection Z -> Z_2
|
|
| 3
|
Groups in Linear Algebra and Complex Variable
|
- Euclidean space as an additive group
- Null space and column space of a linear map
- Invertible linear transformations and matrices
- Determinant
- Additive and multiplicative subgroups of complex numbers
|
- GL(n,R), O(n,R), SL(n,R), SO(n,R)
- C, C*, S^1, n-th roots of unity
|
|
| 15
|
- Catch up and review for final
- Study days
|
See also
