Math 120A: Introduction to Group Theory, Spring 2017
Welcome. Groups come from the natural idea of symmetries that we observe in the real world. Group theory provides both a formalism to work with notions of symmetry in many scientific areas, as well as a great introduction to the methods of abstract algebra.
Here is the syllabus.
Text: A First Course in Abstract Algebra, by J. B. Fraleigh
Assessment consists of:
Weekly homeworks (30%)
In-class problem sessions on Thursdays (10%, graded based on completion, worst two dropped)
Midterm (May 8, 20%)
Final exam (comprehensive) (Jun 12, 40%)
Homeworks
It is crucial for your success in this course that you do homework problems, which will be posted here weekly.
Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Homework 6
Homework 7
Lectures
The notes here are my own notes for the lectures. I am sharing them with you in case they are useful.
| Lecture 1 | Inventing the idea of a group from symmetries | |
| Lecture 2 | Basics: sets and functions recap | |
| Lecture 3 | Cardinality, Equivalence Relations | |
| Lecture 4 | Binary Operations, properties | |
| Lecture 5 | Complex numbers recap, fundamental theorem of algebra | |
| Lecture 6 | Real polynomials, idea of isomorphisms of set-operation pairs | |
| Lecture 7 | Definition of a group, examples, roots of unity | |
| Lecture 8 | More examples of groups | |
| Lecture 9 | First lemmas about groups, subgroups, examples | |
| Lecture 10 | Cyclic groups, subgroup generated by g | |
| Lecture 11 | Dihedral Group D_6 | |
| Lecture 12 | Cyclic groups and subgroups, order | |
| Lecture 13 | Cyclic groups are isomorphic to Z or Z/nZ | |
| Lecture 14 | Subgroups of cyclic groups are cyclic | |
| Lecture 15 | Characterizations of G.C.D. and subgroups of cyclic groups | |
| Lecture 17 | Permutation groups | |
| Lecture 18 | Cycle decomposition | |
| Lecture 19 | Transpositions, odd/even permutations | |
| Lecture 20 | Proof of well-definedness of evenness and oddness | |
| Lecture 21 | Alternating Group, 15-Puzzle, Cayley’s Theorem | |
| Lecture 22 | Cayley’s theorem proof, cosets | |
| Lecture 23 | Cosets and Lagrange’s Theorem and corollaries | |
| Lecture 24 | Group homomorphisms | |
| Lecture 25 | Normal subgroups | |
| Lecture 26 | Quotient groups | |
| Lecture 27 | Examples, First isomorphism theorem | |
| Lecture 28 | Applications of isomorphism theorem | |
| Lecture 29 | Last lecture |
###Office Hours
- Office Location: 510P Rowland Hall (5th floor, turn right when you exit the elevator, then left, 510 is a door on the right past the tutoring center)
- Office hours: Mondays 4-5 pm, Wednesdays 11 am - 12 pm, Fridays 4-5 pm
###Contact Information
- Umut Isik
- Email: misik@uci.edu
- Phone: (949) 824-3153
###Your TA
- Ching-Heng Chiu
- Office hours: Wednesdays 10 am - 12 pm, Thursdays 2 pm - 3 pm
- Office: 540W
###Campus Resources