MATH 3GR3
Abstract Algebra
Fall 2023

Course Information:


Assignments, Tests, and Solutions:




Test Dates (tentative):

There will be two 50-minute tests held during the semester. These tests will be held in-person during the scheduled tutorial time during the weeks of October 17 and November 14.  Each test will be worth 25% of your final grade.

Final Exam Information:

A final exam for this course will be scheduled by the registrar.  Details will be posted on the course website as they become available.


Lecture Schedule (tentative):

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

SageMath:

The following cell can be used to enter and then execute a sequence of SageMath commands.  This can be used to perform calculations for some of the homework questions or just to explore some of the concepts that are covered in this course.  To save the commands and results you can copy and paste them into the document that contains your homework solutions, or you could take a screenshot (or a picture) of the cell (after you have performed the calculation) and include the screenshot file  in your homework solutions document.  Alternatively you can copy and paste the link that is produced when the "share" button is pressed.  This button appears once the "Evaluate" button has been pressed.

For more details on SageMath, consult the online version of the course textbook, or the SageMath website, or the SageCell website. Here is a link to a SageMath tutorial:  https://doc.sagemath.org/html/en/tutorial/





Matt Valeriote