## Mathematics 322, Introduction to algebra, Section 101. September-December 2013, Leonard S. Klinck Building, rm. 460, TuTh 14:00-15:30

• Instructor : Zinovy Reichstein
Office: 1105 Math Annex
Phone: 822-3929
E-mail: reichst at math.ubc.ca

• Textbook : Joseph Gallian, Contemporary Abstract Algebra, 8th edition.
• Course description : Math 322-23 is UBC's undergraduate honours abstract algebra sequence. Math 322 is devoted entirely to group theory, with an emphasis on finite groups. The topics I plan to cover are as follows.

• Preliminaries on integes, congruences, sets, maps and equivalence relations.
• Definition and first properties of a group,
• Cyclic and permutation groups.
• Subgroups, cosets, and Lagrange's Theorem.
• Homomorphisms, normal subgroups, quotients, and simple groups.
• Group actions, p-groups and Sylow theorems.
• Finite abelian groups.

If time permits, I may cover additional topics from the text at the end of the term.

• Evaluation : Homework assignments will be posted on the course website and collected in class. Late homework will not be accepted. The solutions you turn in should be your own, written in your own words. There will be two midterms and a final exam. The midterms are scheduled for Thursday, October 10 and Thursday, November 7. The final exam is scheduled for 7pm on Wednesday, December 18, in BUCH A202. I will compute the total term mark in two ways,

Total 1 := Homework (20%) + Midterm 1 (20%) + Midterm 2 (20%) + Final exam (40%), and

Total 2 := Homework (20%) + Best midterm (20%) + Final exam (60%).

I will use whichever of these two numbers is higher.

• Students with disabilities : Students with documented disabilities who may need special accommodations should make an appointment with me early in the term.
• ### Homework assignments

Problem Set 1. Due in class Thursday, September 19.
Pages 23-27. Problems 4, 11, 14, 16, 18, 26, 36, 50, 54, 62

Problem Set 2. Due in class Tuesday, October 1.
Pages 23-27. Problems 58, 64
Pages 54-58. Problems 22, 30, 36, 50. In problem 50 "multiple of 5" should read "multiple of 4".
Pages 68-75. Problems 4, 16, 48, 80

Problem Set 3. Due in class Tuesday, October 8.
Pages 87-92. Problems 8, 14, 22, 30.
Pages 118-123. Problems 2, 8, 24, 28, 32, 46.

Problem Set 4. Due in class Tuesday, October 29.
Pages 138-142. Problems 2, 4, 12, 36.
Pages 156-160. Problems 10, 12, 26, 34, 38, 48. In Problem 10, HK is defined as { hk | h in H and k in K}.

Problem Set 5. Due in class Tuesday, November 5.
Pages 156-160. Problems 52, 56.
Pages 200-205. Problems 6, 14, 38, 52, 62, 64, 70, 78.
Extra credit Problem: Generalize the assertion of Problem 70 from S_4 to S_n by showing that A_n is the only subgroup of S_n of index 2.

Problem Set 6. Due in class Thursday, November 21.
Pages 421--425. Problems 8, 10, 18, 20, 26, 36, 46, 54

Problem Set 7. Due in class Thursday, November 28.
pp. 234--236. Problems 2, 4, 8, 10, 12, 22, 24, 30.
p. 238--239. Problems 24.
Problem A: Let G = Z_{360} x Z_{300} x Z_{200} x Z_{150}.
(a) Find the elementary divisors of G.
(b) Find the invariant factors of G.
(c) Find the orders of the torsion subgroups G[2], G[3], and G[5].