University of Alberta · Fall Term 2026

MATH 327: Algebra 1

Basic group theory: groups, subgroups, normal subgroups, homomorphisms, quotient groups, coset decomposition, example: permutation group and general linear group; basic (commutative) ring theory: rings, subrings, homomorphisms, ideals, quotient rings, modules over rings, submodules and quotient modules, fraction field; further group theory: groups operating on a set, Sylow theorems.

Instructor: Manish Patnaik
Email: patnaik@ualberta.ca
Office hours: Mondays, 2:00 – 3:00 PM
Location: UCOMM 5-254 (or nearby!)

Announcements

Class announcements

Monday, October 19, 2026

Homework 3 is posted. It covers Lectures 15–19, including product groups in Lecture 18 and quotient groups in Lecture 19. The suggested completion date is Monday, October 26.

Earlier announcements

Tuesday, October 6, 2026

Extra office hours: Wednesday, October 7, 2–3 pm.

Additional short-answer and true/false review questions and solutions are now posted for Midterm 1.

Monday, October 5, 2026

Homework 2 has been revised to remove the questions from Sections 10–12 and the optional semidirect products question. The Homework 2 solutions and updated Midterm 1 coverage sheet are also posted.

Office hours this week are Monday, October 5, 2–3 pm and Wednesday, October 7, 2–3 pm.

Sunday, September 20, 2026

Homework 2 is posted. It covers Lectures 8–13; the suggested completion date is October 5.

Monday, September 14, 2026

added question on permutaitons to homework 1 and updated coverage information for it, lecture 5 posted

Tuesday, September 8, 2026

Lecture 3 & 4 (preliminary) added

Wednesday, September 2, 2026

Lecture 1 updated. Office hours Mondays 2-3pm starting Sep. 14. Email me before if you want to discuss something.

Tuesday, September 1, 2026

Lecture 1 notes posted. Hwk 1 posted.

Tuesday, August 25, 2026

Welcome to MATH 327. Lecture notes will be posted here after each class, and problem sets appear under Problem sets as the term goes on. Note that they are not to be turned in. They are just for your practice.

Office hours will start on the second week of class.

The plan is to use this website for most things, but I will still use Canvas to send out emails to the entire class (as the website isn't synced to the class roster) and to upload your midterm/final grades.

Syllabus

Course information

Meets: MWF, 9:00 – 9:50 AM
Location: CAB 281

Algebra 1 syllabus Simple Syllabus · Fall 2026 View →

Textbook: Michael Artin’s Algebra, 2nd Edition

Grading

Component	Weight
Midterm 1	25%
Midterm 2	25%
Final exam	50%

Schedule & Notes

Lecture schedule

All section numbers refer to M. Artin’s Algebra, 2nd edition.

Lecture	Date	Topic	Notes
1	Wed, Sep 2	Groups and examples, 2.1–2.2	PDF
2	Fri, Sep 4	Basic properties and further examples, 2.2	PDF
3	Wed, Sep 9	Permutations and the symmetric group, 1.5 and 2.2	PDF
4	Fri, Sep 11	Simple transpositions and inversions, 2.2	PDF
5	Mon, Sep 14	Even permutations, S₃	PDF
6	Wed, Sep 16	Subgroups and generators; subgroups of Z, greatest common divisors, and Bézout’s identity, 2.2–2.4	PDF
7	Fri, Sep 18	GCD review; homomorphisms, kernels, images, and isomorphisms, 2.3 and 2.5–2.6	PDF
8	Mon, Sep 21	Automorphisms, conjugation, and normal subgroups, 2.5–2.6; conjugation of permutations, 7.5	PDF
9	Wed, Sep 23	Kernels and normal subgroups; determinant, sign, and the alternating group, 1.4–1.5 and 2.5; introduction to simple groups (overview; see 7.4–7.5)	PDF
10	Fri, Sep 25	Left and right cosets, normality, partitions, and congruence classes modulo 8, 2.7–2.9	PDF
11	Mon, Sep 28	Cosets and partitions; index, Lagrange’s theorem, and groups of prime order, 2.7–2.9	PDF
12	Fri, Oct 2	Review of the counting formula and Lagrange’s theorem; classification of groups of order 6 as cyclic or isomorphic to S₃, 2.8 (see also Ch. 2, Ex. M.3)Class was held on Zoom.	PDF In-class notes (PDF)
13	Mon, Oct 5	Counting formula for homomorphisms and the order of the alternating group; tower law for indices, 2.7–2.9	PDF
14	Wed, Oct 7	Left and right cosets; bijection between left and right cosets; normal subgroups	PDF
—	Fri, Oct 9	Midterm 1	Details →
15	Wed, Oct 14	Correspondence theorem and proof; the homomorphism from S₄ to S₃	PDF
16	Fri, Oct 16	Review of the correspondence theorem, normality, and orders; construction of the homomorphism from S₄ to S₃; application to subgroups of S₄	PDF
17	Mon, Oct 19	Group actions as homomorphisms; the dihedral group D₈ and symmetries of the square	PDF
18	Wed, Oct 21	General dihedral groups; the correspondence theorem for subgroups of S₄; external direct products, 2.11	PDF
19	Fri, Oct 23	Products of cyclic groups; internal direct products and their recognition criterion; quotient groups and the quotient map, 2.11–2.12	PDF
20	Mon, Oct 26	Conjugacy classes in A_n, 7.4	—
21	Wed, Oct 28	Simplicity of A_n	—
22	Fri, Oct 30	p-groups and the first Sylow theorem	—
23	Mon, Nov 2	The second and third Sylow theorems	—
24	Wed, Nov 4	Groups of orders 12, 15, and 21	—
25	Fri, Nov 6	Midterm 2	Details →
26	Mon, Nov 16	Rings and polynomial rings, 11.1–11.2	—
27	Wed, Nov 18	Ring homomorphisms and ideals, 11.3	—
28	Fri, Nov 20	Quotient rings, 11.4	—
29	Mon, Nov 23	First isomorphism and correspondence theorems	—
30	Wed, Nov 25	Adjoining elements	—
31	Fri, Nov 27	Further examples of adjoining elements	—
32	Mon, Nov 30	Factorization, prime elements, and irreducibles, 12.2	—
33	Wed, Dec 2	UFDs; prime versus irreducible, 12.2–12.3	—
34	Fri, Dec 4	Gauss’s lemma; factoring in Z[x] versus Q[x], 12.3	—
35	Mon, Dec 7	Review and catch-up	—

Homework

Problem sets

Practice problems to test your understanding. These are not to be turned in. Please attempt as many as you can!

Set	Covers	Suggested completion date	PDF	Solutions
HW 1	Lectures 1–7	Sep 18	PDF	PDF
HW 2	Lectures 8–13	Oct 5	PDF	PDF
HW 3	Lectures 15–19	Oct 26	PDF	Not yet posted

Assessment

Midterm 1

Held at lecture 15.

Date & time: Friday, October 9, 2026 · In class
Format: In-class, closed book
Coverage: Lectures 1–14 · 2026 coverage sheet (PDF)
2026 exam & solutions: 2026 Midterm 1 (PDF) · 2026 solutions (PDF)
2026 exam statistics: Exam statistics (PDF)
Review materials: 2025 Midterm 1 (PDF) · 2025 solutions (PDF)
Extra practice: Extra practice for Midterm (PDF) · Solutions (PDF)

Assessment

Midterm 2

Held at lecture 26.

Date & time: Friday, November 6, 2026 · In class
Format: In-class, closed book
Coverage: Lectures 16–25
Review materials: Posted closer to the exam

Assessment

Final exam

Cumulative — comprehensive coverage of the full term, weighted most heavily in the final grade.

Date & time: Set by the registrar — TBA
Format: Closed book, comprehensive
Coverage: All course material
Review materials: Posted in the last week of term