MTH 331/431 Abstract Algebra I (Fall 2021):


Schedule:



Week of
Class Topics Remarks, Materials, Assignments

8/30 Introduction
Preliminaries (Ch. 1)
Introduction slides

Sets and Equivalence Relations

9/6 No class on 9/6
(Labor Day)

Integers (Ch. 2)
Induction and Well-Ordering Principle

Division Algorith, Euclidean Algorithm, Prime Numbers

Homework 1 due 9/10:
All students: Ch. 1, §1.4 "Exercises" (online text),
exer. 7, 9, 18(b,c), 19, 20(b), 21, 22(c,e), 25(d).
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 26, 28, and 29.

Latex template for Homework 1: here.
Homework 1 Solutions

9/13 Groups (Ch. 3) Integer Equivalence Classes and Symmetries

Homework 2 due 9/17:
All students: Ch. 2, §2.4 "Exercises" (online text),
exer. 2, 5, 12, 15(d, f), 18.
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 16 and 23.

Latex template for Homework 2: here.
Homework 2 Solutions

9/20 Groups Continued
Subgroups (Ch. 3)

Cyclic Groups (Ch. 4)
Groups: Definitions and Examples

Group examples handout

Subgroups

Homework 3 due 9/24:
All students: Ch.2, §2.4, exer. 31;
Ch. 3, §3.5, exer. 1 (b,d), 2 (c,d), 6, 7, 8, 10, 14.
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 28 in Section 2.4, and 15 and 26 in Section 3.5.

Latex template for homework: here.
Homework 3 Solutions

9/27 Exam I on 9/27 (Monday)

Exam I solutions:
MTH 331 version
MTH 331&/431 version

Cyclic Groups Continued
Cyclic Groups

Subroups of the Circle Group

Study for Exam I: study guide

Homework 4 due 10/1:
All students: Ch. 3, §3.5, exer. 16, 31, 34, 39, 43, 48.
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 46 and 47.
NOTE: bonus problem for all - exer. 45.

Latex template for homework: here.
Homework 4 Solutions

10/4 Permutation Groups. (Ch. 5)
Permutation Groups

Explore Sage sections 1.6, 2.7, 3.8, and 4.8.

Homework 5 due 10/8:
All students: Ch. 4, §4.5,
exer. 1 (a,b,c), 2 (a,b,d,f), 3 (b,c,h,k), 4 (d), 11, 23 (a), 32.
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 1 (e), 23 (c), and 26.
NOTE: bonus problem for all - exer. 31.

Latex template for homework: here.
Homework 5 Solutions

10/11 Permutation Groups Continued.

Dihedral Groups.

Cosets and Lagrange's Theorem.
(Ch. 6)

No class on Friday (Fall Recess)

Dihedral Groups

Homework 6 due 10/18:
All students: Ch. 4, §4.5, exer. 6;
Ch. 5, §5.4, exer. 1, 2 (b,d,g,h,j,p), 3 (b,d), 4, 22, 23, 24, 27.
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 13 and 35 in Section 5.4.
NOTE: bonus problems for all: 6 and 34 in Section 5.4.

Latex template for homework: here.
Homework 6 Solutions

10/18 Cosets and Lagrange's Theorem Cont'd.
Fermat's and Euler's Theorems.

Isomorphisms. (Ch.9)
Cosets and Lagrange's Theorem
Fermat's and Euler's Theorems

Homework 7 due 10/29 (new deadline!):
All students: §6.5, exer. 1, 3, 4, 5 (a,b,c),
11 (only prove statements (a) and (d) are equivalent), 17.
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 18, 19.
NOTE: bonus problems for all: 5 (f), 13.

Latex template for homework: here.
Homework 7 Solutions

10/25 Exam II on 10/25 (Monday)

Exam II solutions:
MTH 331 version
MTH 331&/431 version

Isomorphisms Cont'd.
Study for Exam II: study guide .

Isomorphisms

Topics for presentations

Homework 8 due 11/8:
All students: §9.4, exer. 2, 7, 8,
9 (do not need to proof that G is a group, prove isomorphism only),
16 (b and d), 34, 46, 50.
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 24 and 41.
NOTE: bonus problems for all: 15, 25.

Latex template for homework: here.
Homework 8 Solutions

1/11 Direct Products. (Ch.9)

Normal Subgroups and
Factor Groups. (Ch.10)
Direct Products

11/8 Factor Groups Continued.

Homomorphisms. (Ch.11)
Normal Subgroups. Factor Groups. Simple Groups.

Homomorphisms

Homework 9 is due 11/15:
All students: §10.4, exer. 1 (a,b,c,e), 5, 6, 8, 13 (a,c).
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 9, 11.
NOTE: bonus problems for all: 13 (b,d).

Latex template for homework: here.
Homework 9 Solutions

11/15 The Isomorphisms Theorems. (Ch.11)

Finite Abelian Groups (briefly).
(§13.1 - first part)

Rings. (Ch.16)

The Isomorphisms Theorems

Finite Abelian Groups

Homework 10 is due 11/30:
All students: §11.4, exer. 2 (b,c,d,e), 4, 9, 10, 12, 13 and
§13.4, exer. 2 and 3.
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 16 and 18 in §11.4.
NOTE: bonus problems for all: exer. 7 (a,b,c) in § 11.4.

Latex template for homework: here.

11/22 Exam III on 11/22 (Monday)

Exam III solutions:
MTH 331 version
MTH 331&/431 version

Tuesday follows Thursday schedule,
no class!

Thanksgiving Break: 11/24-11/28
Study for Exam III: study guide

11/29 Rings Continued. Integral Domains and Fields. (Ch.16)

Rings

Integral Domains and Fields

Ring examples handout

Homework 11 is due 12/10:
All students: §16.7, exer. 1 (b,e), 3 (b,d), 8, 9, 18 (c,d), 23.
Honors (MTH 331&H) and graduate (MTH 431) students:
in addition, do exer. 2 and 31.
NOTE: bonus problems for all: exer. 1(h) and 36.

Latex template for homework: here.