| Jan. 22 |
10-14, 16-19 |
1, 2 : Intro to Groups |
binary operation, group, identity, inverse, general linear group of degree 2 |
| Jan. 24 |
25-30, 34 |
3 : Basic Properties of Groups |
abelian group, order of a group |
| Jan. 27 |
4, 6-7, 20-21, 35- 37 |
0, 2, 4 : Integers |
a divides b, prime, greatest common divisor, relatively prime, least common multiple (see exercise 4.31) |
| Jan. 29 |
33-34,38
| 4 : Order of an Element |
order of an element, element of finite order, element of infinite order |
| Jan. 31 |
39-40 |
4 : Cyclic Groups |
cyclic group, generator of a group |
| Feb. 3 |
notes*, 74-76 |
8, Problem #8.15 : Dihedral Group |
plane symmetry, regular polygon, dihedral group of order 2n (all on attached notes) |
| Feb. 5 |
43-50 |
5: Subgroups |
subgroup, proper subgroup, trivial subgroup, subgroup lattice |
| Feb. 7 |
50-52, 59-62 |
5, 7: Cyclic Subgroups / Functions |
function, onto, one-to-one, domain, range, bijection, inverse function |
| Feb. 10 |
63-64, 66-69 |
8 : Symmetric Groups |
cycles, disjoint cycles, symmetric group on X, transposition |
| Feb. 12 |
70-73 |
8 : Alternating Group |
even permutation, odd permutation, alternating group of degree n |
| Feb. 14 |
81-83 |
9: Equivalence Relations and Z/nZ |
equivalence relation, equivalence class |
| Feb. 17 |
21-22 |
2: Z/nZ, U(n) |
remainder of a mod n, congruent modulo n, additive group of integers mod n |
| Feb. 19 |
83-85 |
9: Cosets |
right coset of H in G, left coset |
| Feb. 21 |
83-85, 88-91 |
9, 10 : Cosets, Lagrange's Theorem |
Lagrange's Theorem, cardinality, index, coset representative (see 82) |
| Feb. 24 |
92,103-104 (assuming abelian) |
11 : Quotient Group |
quotient of abelian groups |
| Feb. 26 |
99-102 |
11: Normal Subgroups, Quotient Groups |
normal subgroups, quotient groups |
| Feb. 28 |
103-105, 55-57 |
11 : Cauchy's Theorem and other Consequences |
|
| Mar. 2 |
55-57, 109-113 |
6,12 : Direct Product, Homomorphisms |
direct product, homomorphism, isomorphism, monomorphism, epimorphism, automorphism |
| Mar. 4 |
Exam 1 |
1-5, 7-10 |
|
| Mar. 6 |
114-115, 121-122 |
12, 13 : Homomorphism Properties |
inverse image, homomorphic image |
| Mar. 9 |
122-124, 127 |
13: Isomorphism Theorems |
kernel, fundamental theorem on gp homomorphisms, |
| Mar. 11 |
125-126, 128-130, 153-154 |
13, 16 : Isom. Theorems, Rings |
second and third isomorphism theorems, ring, commutative ring, unity |
| Mar. 13 |
155-160 |
16: Ring Properties, Integral Domains |
zero divisor, unit , nilpotent, trivial ring, integral domain, division ring, field |
| Mar. 30/31 |
164-170 |
17: Subrings, Ideals, Prime Ideals |
subring, ideal, proper ideal, trivial ideal, principal ideal generated by a |
| Apr. 1/2 |
171-182 |
17: Maximal Ideals, 18: Ring Homomorphisms |
ring homomorphisms, canonical homomorphism, characteristic 0, characteristic p |
| Apr. 6/7 |
183-193 |
18: Quotient Fields |
quotient field of an integral domain, extension, polynomial rings, leading coefficient, constant polynomial |
| Apr. 8/9 |
194-198 |
19: Factoring |
root, irreducible, primitive |
| Apr. 13/14 |
199-201 |
19: Irreducibility Tests |
Gauss' Lemma, Eisenstein's Criterion, mod p Irreducibility (Thm 19.12) |
| Apr. 15/16 |
205-207 |
20: Ideals in Polynomial Rings |
recap prinicpal ideals and maximal ideals |
| Apr. 20/21 |
211-212, 217-218, 220-221 |
21: Principal Ideal Domain |
prime, irreducible, associates, principal ideal domain, Euclidean domain |
| Apr. 22/23 |
-- |
Flexible Day/Exam Prep |
|
| Apr. 24 |
Exam 2 [instructions] |
6, 11-13, 16-19 |
|
| Apr. 27/28 |
213-216, 218-220 |
21: Unique Factorization Domain |
unique factorization domains |
| Apr. 29/30 |
93-94, 98 (10.32), Notes on Pweb: 133-139 |
Introduction to Group Actions |
centralizer, conjugacy classes, group acting on a set, orbit, stabilizer, not in book: faithful action, kernel, transitive action |
| May 4/5 |
95, Notes on PWeb: 140-144 |
The Class Equation and Consequences |
|
| May 6/7 |
Notes on PWeb: 144-148 |
Simplicity of A5 |
|