Math 415H Homework

Homework Assignment 1: due Wednesday, 9/4
Section 4: 4, 5, 13, 16, 18, 19, 29, 32
5: 8, 10, 13, 20, 23, 34, 36, 51, 54

Homework Assignment 2: due Wednesday, 9/11
1: 6, 9, 12, 20, 26
2: 23, 34, 36, 37
3: 8, 10

In addition, complete the following exercise based on Persi Diaconis' lecture:
A. Let G be the integers, a group under addition. Let H be the subgroup of all multiples of 3. Let X be the standard set of coset representatives: 0, 1, 2.
(1) Find C(X). (Recall that C(X) is the ratio of the number of pairs x,y from X for which x+y is in X to the square of the cardinality of X.)
(2) Is there a choice of coset representatives Y for which C(Y) is larger than C(X)? If so, find one, and compute C(Y).

Bonus (optional). Let Z and Q denote the groups of integers and of rational numbers under addition, respectively. Are Z and Q isomorphic? Justify your answer.

Homework Assignment 3: due Wednesday, 9/18
6: 22, 38, 39, 45, 46, 49, 52
7: 2, 16, 18
8: 1, 3, 16, 18, 20, 21b, 22

Homework Assignment 4: due Wednesday, 9/25
9: 1, 5, 13, 33
10: 4, 15, 26, 34

EXAM I: Wednesday 9/25

Homework Assignment 5: due Wednesday, 10/9
10: 45, 46, 47
11: 3, 16, 21, 39, 42, 46, 50, 51
12: 1c,d, 8, 10, 18

Homework Assignment 6: due Wednesday, 10/16
13: 8, 13, 19, 22, 33, 36, 38, 47, 50, 55
14: 6, 10, 30, 31, 32, 33, 40

Homework Assignment 7: due Wednesday, 10/23

Homework Assignment 8: due Wednesday, 10/30