MTH 331/431 Abstract Algebra I (Fall 2019):


Schedule:

Week of
Class Topics Remarks, Materials, Assignments

8/25 Introduction.
Preliminaries. (Ch. 1)
Introduction notes
Sets and Equivalence Relations

9/1 No class on 9/2
(Labor Day)

Integers. (Ch. 2)
Induction and Well-Ordering Principle
Division Algorith, Euclidean Algorithm, Prime Numbers

Homework 1 due 9/6:
All students: Ch. 1, §1.3 "Exercises" (online text),
exer. 7, 9, 18(b,c), 19, 20(b), 21, 22(c,e), 25(d).
Graduate (MTH 431) students: in addition, do exer. 26 and 29.
(MTH 331 students are welcome to do these problems for extra credit.)
Latex template for homework: here.

Homework 1 Solutions

9/8 Groups. (Ch. 3) Integer Equivalence Classes and Symmetries

Homework 2 due 9/13:
All students: Ch. 2, §2.3 "Exercises" (online text),
exer. 2, 5, 12, 15(d, f), 18.
Graduate (MTH 431) students: in addition, do exer. 17(c) and 23.
(MTH 331 students are welcome to do these problems for extra credit.)
Latex template for homework: here.

Homework 2 Solutions

9/15 Groups Cont'd.
Subgroups. (Ch. 3)
Groups: Definitions and Examples
Summary of Some Groups Examples

Subgroups

Homework 3 due 9/20:
All students: Ch.2, §2.3, exer. 28, 31;
Ch. 3, §3.4, exer. 1 (b,d), 2 (c,d), 6, 7, 8, 14.
Graduate (MTH 431) students: in addition, do exer. 10 and 15 in Section 3.4.
(MTH 331 students are welcome to do these problems for extra credit.)
Latex template for homework: here.

Homework 3 Solutions

9/22 Exam I on 9/23 (Monday)

Exam I solutions:
MTH 331 version
MTH 431 version

Cyclic Groups. (Ch. 4)
Cyclic Groups

Study for Exam I (9/23): study guide

Homework 4 due 9/27:
All students: Ch. 3, §3.4, exer. 16, 26, 33, 39, 43, 48.
Graduate (MTH 431) students: in addition, do exer. 34 and 47.
(MTH 331 students are welcome to do these problems for extra credit.)
NOTE: Extra problem for all: 45.
Latex template for homework: here.

Homework 4 Solutions

Sage Lab on Thursday: explore "Sage" Sections 1.5, 2.6, and 3.7.

9/29 Cyclic Groups Cont'd.
Permutation Groups. (Ch. 5)
Subroups of the Circle Group

Homework 5 due 10/7 (Monday):
All students: Ch. 4, §4.4, exer. 1 (a,b,c), 2 (a,b,d), 3 (b,c,h,k), 4 (d), 11, 23 (a), 31.
Graduate (MTH 431) students:
in addition, do exer. 2 (f), 23 (c), and 26.
(MTH 331 students are welcome to do these problems for extra credit.)
NOTE: Extra problem for all: 32.
Latex template for homework: here.

Homework 5 Solutions

10/6 Permutation Groups Cont'd.

Fall Recess 10/10 and 10/11:
no classes!

Permutation Groups
Dihedral Groups

Homework 6 due 10/16:
All students: Ch. 4, §4.4, exer. 6;
Ch. 5, §5.3, exer.1, 2 (b,d,g,h,j,p), 3 (b,d), 4, 22, 23, 24, 27.
Graduate (MTH 431) students:
in addition, do exer. 13 in Section 5.3.
(MTH 331 students are welcome to do these problems for extra credit.)
Extra problems for all: 6 and 34 in Section 5.3.
Latex template for homework: here.

Homework 6 Solutions

10/13 Exam II on 10/17 (Thursday)

Exam II solutions:
MTH 331 version
MTH 431 version

Cosets and Lagrange's Theorem
(Ch. 6)
Study for Exam II (10/17): study guide

Homework 7 due 10/25:
All students: §6.4, exer. 1, 3, 4, 5 (a,b,c), 8,
11 (only prove statements (a) and (d) are equivalent), 17.
Graduate (MTH 431) students: in addition, do exer. 18, 19.
(MTH 331 students are welcome to do these problems for extra credit.)
Extra problems for all: 5 (f), 13.
Latex template for homework: here.

Homework 7 Solutions

10/20 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

10/27 Isomorphisms Cont'd.
Direct Products (Ch.9)
Isomorphisms: Definitions and Examples
Direct Products

Homework 8 due 11/6:
All students: §9.3, exer. 2, 3, 7, 8,
9 (do not need to proof that G is a group, prove isomorphism only),
16 (b and d), 34, 46, 50.
Graduate (MTH 431) students: in addition, do exer. 24 and 41.
(MTH 331 students are welcome to do these problems for extra credit.)
Extra problems for all: 15, 25.
Latex template for homework: here.

Homework 8 Solutions

11/3 Normal Subgroups and Factor Groups. (Ch.10) Homomorphisms (Ch.11) Normal Subgroups. Factor Groups. Simple Groups.

Homework 9 is due 11/13:
All students: §10.3, exer. 1 (a,b,c,e), 5, 6, 8, 9, 13 (a,c).
Graduate (MTH 431) students: in addition, do exer. 11 and 12.
(MTH 331 students are welcome to do these problems for extra credit.)
Extra problems for all: 13 (b,d).

11/10 Homomorphisms Cont'd. Study for Exam III (11/14): study guide

11/17

11/24 Tuesday follows Thursday schedule

Thanksgiving Recess:
no classes 11/27-29

12/1 Student presentations: list of topics

12/8 Monday follows Friday schedule

Final Exams start 12/10/19