Lecture # | Date | topics covered | reading/resources/comments |
1 | 05/09/23 | Course introduction, equivalence relations | section 1.2 from textbook |
2 | 07/09/23 | equivalence relations, partitions | section 1.2 |
3 | 8/09/23 | partitions, division algorithm | sections 1.2, 2.2 |
4 | 12/09/23 | gcd(a, b), Euclidean algorithm, symmetries of the rectangle | sections 1.6, 3.1 |
5 | 14/09/23 | Cayley table of symmetry groups of rectangle and triangle, definition of a group |
sections 3.1, 3.2 |
6 | 15/09/23 | definition of a group, examples, basic properties: the
cancellation law for groups, properties of the Cayley table, exponentiation laws |
section 3.2 |
7 | 19/09/23 | group of integers modulo n, group of units modulo n | section 3.2 |
8 | 21/09/23 | subgroups, examples, Proposition 3.30 | sections 3.2, 3.3 |
9 | 22/09/23 | examples of subgroups, cyclic subgroups, and cyclic groups. Assignment #1 due. | sections 3.3, 4.1 |
10 | 26/09/23 | cyclic groups, order of groups, elements. Cyclic groups are abelian, every subgroup of a cyclic group is cyclic | sections 4.1 Some notes |
11 | 28/09/23 | cyclic groups, orders of elements in a cyclic group, permutation groups | sections 4.1, 5.1 |
12 | 29/09/23 | permutation groups, examples, cycles | section 5.1 |
13 | 03/10/23 | cycle decomposition of permutations, transpositions, examples | section 5.1 |
14 | 05/10/23 | even and odd permutations, the alternating groups | section 5.1 |
15 | 06/10/23 | Lemma 5.14, dihedral groups. Assignment #2 due. | section 5.2 |
16 | 17/10/23 | Midterm test review. Midterm
test #1 (during tutorial). |
sections 5.2, 6.1 |
17 | 19/10/23 | left and right cosets of a subgroup | section 6.1 |
18 | 20/10/23 | cosets, Lagrange's Theorem | sections 6.1, 6.2 |
19 | 24/10/23 | Lagrange's Theorem, Fermat's Little Theorem, Euler's Theorem | sections 6.2, 6.3 |
20 | 26/10/23 | isomorphisms | section 9.1 |
21 | 27/10/23 | isomorphisms, cyclic groups, Cayley's Theorem. Assignment #3 due. | section 9.1 |
22 | 31/10/23 | Cayley's Theorem, Cartesian Products | sections 9.1, 9.2 |
23 | 02/11/23 | Cartesian product, product of cyclic groups, internal direct product. | section 9.2 |
24 | 03/11/23 | internal direct product, normal subgroups | sections 9.2, 10.1 |
25 | 07/11/23 | factor groups, examples. | sections 10.1, 10.2 |
26 | 09/11/23 | simple groups, homomorphisms, examples, properties | section 11.1 Lecture Notes |
27 | 10/11/23 | the kernel, properties of the kernel, canonical homomorphism, First Isomorphism Theorem. Assignment #4 due. | sections 11.1, 11.2 Lecture Notes |
28 | 14/11/23 | midterm test review. Midterm test #2 (during tutorial). | |
29 | 16/11/23 | the first isomorphism theorem and applications. | section 11.2 Lecture Notes |
30 | 27/11/23 | Rings, ring examples, basic properties | section 16.1 |
31 | 21/11/23 | integral domains, ring homomorphisms | sections 16.2, 16.3 |
32 | 23/11/23 | homomorphisms, kernels, ideals. | section 16.3 |
33 | 24/11/23 | ideals, continued. Assignment #5 due. | section 16.3 |
34 | 28/11/23 | factor rings, canonical homomorphisms, First Isomorphism Theorem. | section 16.3 |
35 | 30/11/23 | maximal and prime ideals | section 16.4 |
36 | 01/12/23 | polynomial rings, the division algorithm | sections 17.1, 17.2 |
37 | 05/12/23 | the division algorithm. Assignment #6 due. | section 17.2 |