Math 3140 Spring 24
## MATH 3140: Abstract Algebra 1 (Spring 2024)

Syllabus
### Office hours

M 12:30 - 2:00 pm, W 10 - 11 am at Math 310
### Final Exam:

Tuesday 05/07, 1:30 - 4 pm, in class
### Schedule

Numbers are for orientation and refer to sections with related material in
the book by Gallian.
- 01/17: symmetries, multiplication tables (1)
- 01/19: groups, basic properties (2)
- 01/22: subgroups, center (3)
- 01/24: review integers mod n (see handout)
- 01/26: order of elements (3), cyclic groups (4)
- 01/29: subgroups of cyclic groups (4)
- 01/31: permutation groups, Sym X, S_n (5)
- 02/02: homomorphisms (10), isomorphic groups (6)
- 02/05 automorphisms (6)
- 02/07: (inner) automorphism groups, Cayley's Theorem (6)
- 02/09: cosets, index of subgroups, Lagrange's Theorem (7)
- 02/12: applications of Lagrange's Theorem, groups of order p (7)
- 02/14: groups of order 2p (7)
- 02/16: orbit-stabilizer theorem, symmetry group of the cube (7)
- 02/19: REVIEW
- 02/21: MIDTERM 1
- 02/23: discussion of midterm, direct products of groups (8)
- 02/26: orders in direct products (8)
- 02/28: conjugacy, normal subgroups (9)
- 03/01: quotient groups (9), first isomorphism theorem (10)
- 03/04: sign of permutations, alternating groups (5)
- 03/06: correspondence theorem (10)
- 03/08: finite abelian groups, factorization into p-groups (11)
- 03/11: finite abelian groups, factorization of p-groups (11)
- 03/13: group actions (29)
- 03/18: orbits, fixed points, Burnside-Frobenius Lemma (29)
- 03/20: counting orbits
- 03/22: Rubik's cube in GAP, [GAP calculations],
[pdf], hardness of finding shortest solutions [arxiv]
- 04/01: conjugation as group action, class equation, p-groups (24)
- 04/03: groups of size p^2, Sylow subgroups (24)
- 04/05: REVIEW
- 04/08: MIDTERM 2
- 04/10: Sylow Theorems (24)
- 04/12: discussion of midterm
- 04/15: groups of size pq (24)
- 04/17: simple groups, composition series, A_5 is simple (25)
- 04/19: rings (with 1), polynomials, subrings (12)
- 04/22: integral domains, units, fields (13), ring homomorphisms

### Assignments

- due 01/24 [pdf] [tex] [solutions]
- due 01/31 [pdf] [tex] [solutions]
- due 02/07 [pdf] [tex] [solutions]
- due 02/14 [pdf] [tex] [solutions]
- due 02/21 [pdf] [tex] [solutions]
- due 02/28 [pdf] [tex] [solutions]
- due 03/06 [pdf] [tex] [solutions]
- due 03/13 [pdf] [tex] [solutions]
- due 03/22 (updated) [pdf] [tex] [solutions]
- due 04/05 [pdf] [tex] [solutions]
- due 04/17 [pdf] [tex] [solutions]
- due 04/26 [pdf] [tex]

### Handouts

- Equivalence relations [pdf]
- Integers mod n [pdf]
- Basic definitions in group theory [pdf]

### Reading

- Joseph A. Gallian. Contemporary Abstract Algebra. (any edition, available electronically via the CU library)