MATH 420, ALGEBRA I

DATE: Fall Semester, 1995

INSTRUCTOR: John Beachy, Watson 355, 753-6753

OFFICE HOURS: MWF 11:00-12:00, or by appointment.

TEXT: Abstract Algebra with a Concrete Introduction, Beachy/Blair

SYLLABUS: Chapter One, Integers; Chapter Two, Functions; Chapter Three, Groups (3.1-3.6)

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, October 20. The final exam is scheduled for Friday, December 15, 3:00-4:50 p.m.

SCHEDULE OF LECTURES (REVISED):

Monday    Wednesday Friday         S  M Tu  W Th  F  S
                                                      
Intro      1.1       1.1     Aug  27 28 29 30 31  1  2
Holiday    1.1       1.2     Sep   3  4  5  6  7  8  9
 1.2       1.2       1.3          10 11 12 13 14 15 16
 1.3       1.3       1.4          17 18 19 20 21 22 23
 1.4       1.4      EXAM I        24 25 26 27 28 29 30
 1.4       2.1       2.1     Oct   1  2  3  4  5  6  7
 2.1       2.2       2.2           8  9 10 11 12 13 14
 2.2       2.2       2.3          15 16 17 18 19 20 21
 2.3       2.3       3.1          22 23 24 25 26 27 28
 3.1      EXAM II    3.1     Nov  29 30 31  1  2  3  4
 3.2       3.2       3.2           5  6  7  8  9 10 11
 3.3       3.3       3.4          12 13 14 15 16 17 18
 3.4       3.4      Holiday       19 20 21 22 23 24 25
 3.5       3.5      EXAM III      26 27 28 29 30  1  2
 3.6       3.6       3.6     Dec   3  4  5  6  7  8  9
                    FINAL         10 11 12 13 14 15 16

ASSIGNMENTS**

Due Friday, 9/8/95: Section 1.1, p.12: #3g, 3h, 4g, 4h, 5, 9, 13, 17
Due Wednesday, 9/13/95: Section 1.2, p.20: #2, 3, 7, 10, 18
Due Monday, 9/25/95: Section 1.3, p.27: #4, 5, 7, 18, 19
Due Wednesday, 9/27/95: Section 1.4, p.36: #6, 7, 8, 9
Due Wednesday, 10/11/95: Section 2.1, p.59: #6, 7, 8, 9, 12
Due Monday, 10/16/95: Additional problems listed below.
Due Wednesday, 10/25/95: Section 2.2, p.66: #1, 2 c,d and Section 2.3, p.78: #1, 3, 5, 11, 12
Due Wednesday, 11/8/95: Section 3.1, p.99: #6,8,9,20,21
Due Friday, 11/17/95: Section 3.2, p.109: #5,6,12,13,14a-c,14d

The last two assignments will be photocopied and included in the portfolios of undergraduate math majors. The assignments are found below.
Due Friday, 12/1/95: Five questions on 3.3, 3.4, including p.125 #11
Due Friday, 12/8/95: Five questions on 3.5, including p.130 #15, 19

MATH 420        HOMEWORK        10/9/95

Due Monday, October 16

1. Solve the congruence  24x  \congruent 168 mod(200).

2. Solve the congruence  42x  \congruent  12 mod(90).

3. Solve the system of congruences
                x  \congruent 2 mod(9)
                x  \congruent 4 mod(10) .

4. Solve the system of congruences
               5x  \congruent 14 mod(17)
               3x  \congruent  2 mod(13) .

5. In Z_{20}:
find all units (list the multiplicative inverse of each);
find all idempotent elements;
find all nilpotent elements.

6. In Z_{24}:
find all units (list the multiplicative inverse of each);
find all idempotent elements;
find all nilpotent elements.


J. Beachy, E. Behr  MATH 420 Assignments   11/27/95

In addition to the usual grading,
these assignments will be used for assessment purposes.
If you are an undergraduate math major,
after the assignment is graded
a photocopy of your solutions
will be placed in your assessment portfolio.


Due Friday, December 1:


(1) Let G be any group, and let a be a fixed element of G.
Define a function phi_a : G -> G by
phi_a (x) = a x a^{-1}, for all x in G.
Show that phi_a is an isomorphism.
(Exercise 11 of Section 3.4)


(2) Let G be a group, let H be a subgroup of G, and let a belong to G.

(a) Prove that aHa^{-1}
= { x in G | x = aha^{-1}  for some h in H }
is a subgroup of G.

(b) Use the previous exercise to prove that the group aHa^{-1}
is isomorphic to H.


(3) Show that Z_17^x is isomorphic to Z_16.

Hint: Find a generator for Z_17^x,
and use the technique of Example 3.4.1.


(4) For any prime number q,
let GL_2 (Z_q) be the set of all 2 x 2 invertible matrices
with entries from Z_q.

S_2 (Z_q) = {
   _            _
  |              |
  |  a_11  a_12  |
  |              |
  |  a_21  a_22  |
  |_            _|

in GL_2 (Z_q) such that

a_11 + a_21 = 1  and  a_12 + a_22 = 1  }

is called the stochastic group of degree 2 over Z_q.
Prove that S_2 (Z_q) is a subgroup of GL_2 (Z_q).


(5) Let q be a prime number.
The affine group of degree 2 over Z_q
is the subgroup of GL_2 (Z_q) defined as

A_2 (Z_q) = {
   _            _
  |              |
  |  a_11  a_12  |
  |              |
  |   0     1    |
  |_            _|

in GL_2 (Z_q) such that a_11 is nonzero }.

Use Exercise 1, with a =

   _          _
  |            |
  |   1    0   |
  |            |
  |   1    1   |
  |_          _|

to show that S_2 (Z_q) is isomorphic to A_2 (Z_q).


Due Friday, December 8:


(1) Give the lattice diagram of subgroups of Z_{100}.


(2) Let G = A_2 (Z_5),
the affine group of degree 2 over Z_5,
defined in Exercise 5 of the previous exercise set.

(a) In G, find a cyclic subgroup of order 5
and a cyclic subgroup of order 4.

(b) Find the exponent of G.  (See Definition 3.5.6.)


(3) Let G be any group with no proper, nontrivial subgroups,
and assume that | G | > 1.

(a) Show that G must be cyclic.

(b) Show that G cannot be infinite.

(c) Show that G must have prime order.
(Exercise 15, Section 3.5.)


(4) Let p be a prime number,
and assume that G is a group with p^k elements,
where k is a positive integer.
Prove that G has a subgroup of order p.
(Exercise 19, Section 3.5.)


(5) Suppose that p is a prime number of the form p = 2^n + 1.

(a) Show that in Z_p^p the order of [2]_p is 2n.

(b) Use part (a) to prove that n must be a power of 2.


** The above assignments are from the first edition of the book. See the list below for the corresponding problems in the second edition.

1.1: #4b, 4c, 6b, 6c, 7, 11, 15, 19
1.2: #3, 5, 9, 12, 20
1.3: #4, 6, 7, 19, 20
1.4: #7, 8, 9, 10
2.1: #8, 9, 10, 11, 14
2.2: #1, 2 c,d
2.3: #1, 3, 5, 11, 12
3.1: #5, 8, 9, 20, 21
3.2: #5, 6, 12, 14, 16, 17
3.4: #13
3.5: #15, 19


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