### MATH 420, ALGEBRA I

DATE: Spring Semester, 1997

INSTRUCTOR: John Beachy, Watson 355, 753-6753

OFFICE HOURS: MWF 10:00-10:50; MW 1:00:2:00, or by appointment.

TEXT: Abstract Algebra, Second Edition, Beachy/Blair

SYLLABUS: Chapter One, Integers; Chapter Two, Functions; Chapter Three, Groups

COURSE OBJECTIVES: The student is expected to acquire an understanding of the elementary theory of groups, together with the necessary number theoretic prerequisites. There will be some discussion of the computational aspects of these topics, but the main thrust of the course will be theoretical. The student will be expected not only to follow the proofs presented in class and in the text, but also to learn to construct new proofs. Proofs must be logically correct and care must be taken to write them precisely and in grammatically correct English.

COURSE PREREQUISITE: MATH 240, Linear Algebra. We will use matrices in some important examples, but the main reason for the requirement is to attempt to guarantee a certain level of "mathematical maturity."

GRADING: Final grades will be based on 600 points: 3 hour tests (300), homework (100), and a comprehensive final exam (200).
The homework problems are extremely important. In many ways the course is like an English composition course, since it requires you to write out very carefully the reasons for each step in your solutions of problems.
Note that the last day to withdraw from the course without penalty is Friday, March 7.

FINAL: The final exam is scheduled for Wednesday, May 7, 10:00-11:50 a.m.
A review session will be held Monday evening at 7:00 pm in DU 402.

Review material: Previous finals | Chapter summaries

ASSIGNMENTS:

```Monday, 5/5/97:          Three problems for assessment (see below)

Friday, 4/25/97:         Test III, covering 3.2-3.5

Friday, 4/25,97:         Section 3.4: #13, 14
Section 3.5: #3, 10, 11, 16

Friday, 4/18/97:         Section 3.3: #8, 9, 10
Section 3.4: #2, 4, 8

Friday, 4/11/97:         Three problems for assessment (see below)

Friday, 4/4/97:          Test II, covering sections 2.1-2.3, 3.1-3.2

Due Monday, 3/31/97:     Section 3.1, p. 90: #6, 8, 9, 19, 20
Section 3.2, p.101: #3, 4

Due Wednesday, 3/26/97:  Section 2.3, p. 76: #7, 10, 11, 12

Due Wednesday, 3/19/97:  Section 2.2, p. 62: #1, 2, 7, 9, 10

Due Wednesday, 2/26/97:  Section 2.1, p. 54: #2, 5, 7(a-d), 8 - 11, 19

Friday, 2/14/97:         Test I, covering Chapter 1

Due Wednesday, 2/12/97:  Section 1.4, p. 40: #2, 7, 10, 11, 12

Due Friday, 2/7/97:      Section 1.3, p. 30: #3, 5, 7, 9, 13, 15, 18

Due Friday, 1/31/97:     Section 1.2, p. 21: #6, 10, 11, 12, 22

Due Friday, 1/24/97:     Section 1.1, p. 14: #4a,b,c; 6a,b,c; 10, 11, 13, 18

```

SCHEDULE OF LECTURES:

```       MONDAY  WEDNESDAY   FRIDAY          S  M Tu  W Th  F  S
Week of                               JANUARY 1997
1/13   1.1       1.1        1.1          12 13 14 15 16 17 18
1/20  HOLIDAY    1.2        1.2          19 20 21 22 23 24 25
1/27   1.2       1.3        1.3          26 27 28 29 30 31  1
2/3    1.3       1.4        1.4      FEB  2  3  4  5  6  7  8
2/10   1.4       2.1       TEST I         9 10 11 12 13 14 15
2/17   2.1       2.1        2.2          16 17 18 19 20 21 22
2/24   2.2       2.2        2.3          23 24 25 26 27 28  1
3/3    2.3       2.3        3.1      MAR  2  3  4  5  6  7  8
SPRING BREAK               9 10 11 12 13 14 15
3/17   3.1       3.1        3.2          16 17 18 19 20 21 22
3/24   3.2       3.2       TEST II       23 24 25 26 27 28 29
3/31   3.3       3.3        3.3      APR 30 31  1  2  3  4  5
4/7    3.4       3.4        3.4           6  7  8  9 10 11 12
4/14   3.5       3.5        3.5          13 14 15 16 17 18 19
4/21   3.5      TEST III    3.6          20 21 22 23 24 25 26
4/28   3.6       3.6   READING DAY   MAY 27 28 29 30  1  2  3
5/5             FINAL                     4  5  6  7  8  9 10
FINAL EXAM   Wednesday 5/7/97 10:00-11:50 a.m.
```

Assessment 1, due 4/11/97

1. (10 pts) Let G be an abelian group. Let H = { g in G | g = x2 for some x in G }. Prove that H is a subgroup of G.

2. (10 pts) Let G be an abelian group, and let K be a subgroup of G. Let H = { g in G | gn is in K for some n in Z }. Prove that H is a subgroup of G.

3. (10 pts) In GL2(R), define

```     _       _                _       _
|         |              |         |
|  0  -1  |              |  0   1  |
A = |         |     and  B = |         |
|  1   0  |              | -1  -1  |
|_       _|              |_       _|
```
Show that A has order 4, and B has order 3, but the product AB has infinite order.

Assessment 2, due 5/5/97

1. (10 pts) Let G be a group, and let a be a fixed element of G. The set of all elements that commute with a is called the centralizer of a, and is denoted by C(a). In symbols,
C(a) = { x in G | xa=ax } = { x in G | x = axa-1 }.
(a) Show that C(a) is a subgroup of G.
(b) Show that < a > is a subset of C(a).

2. (10 pts) (a) Compute the centralizer of the matrix

```     _       _
|         |
|  1   1  |
A = |         |
|  0   1  |
|_       _|
```
in the general linear group GL2(R).
(b) Compute the centralizer of A in the subgroup of GL2(R) defined by all matrices of the form
```     _       _
|         |
|  m   b  |
|         |
|  0   1  |
|_       _|
```
3. (10 pts) Let G be the dihedral group Dn, where n >= 3.
Let G = { aibj | 0 <= i < n and 0 <= j < 2 }, where a has order n, b has order 2, and ba = a-1b.
(a) Show that C(a) = < a >.
(b) Show that if n is odd, then C(b) = < b >, but that if n is even, say n = 2m, then
C(b) = { e, b, am, amb }.

Other resources:
The WEB site Understanding Mathematics: a study guide has a good discussion about learning mathematics.