Math 120 Problem Set 8 (due Wed, May 29):

Chapter 5 contains considerably more material than we covered in class (or that you need to know for math 120). You should read 5.1 and understand the statement of the fundamental theorem in 5.2 (I hope we'll have time for the proof before the end of the quarter). We didn't talk about 5.3 at all. From 5.4, we only covered theorem 9. From 5.5, you should understand what a semidirect product is, and know theorem 12.

Text problems:
7.1: 11, 12, 14

Non-text problems to turn in:

1. Suppose p and q are distinct primes. Suppose G is an abelian group with p2q elements and that G is NOT cyclic.
(a) What is the largest order of any of the elements of G?
(b) How many elements of each order are there in G?
Hint: use the fundamental theorem for finitely generated abelian groups.

2. Let R= be a cyclic group of order 2. Let G be any group. Suppose we try to make a group S that "dihedralizes" G. The elements of S are the elements of GxR, but the multiplication is:

(x,1)(y,r j) = (xy,r j),
(x,r)(y,r j) = (xy -1,r j+1).

(a) Show that if G is not abelian, then S is a not a group.
(b) Show that if G is abelian, then S is a group.

3. Suppose F is a finite field.
(a) Let p be the order of 1 as an element of the additive group F. (In other words, let p be the smallest positive integer such that p1 = 0, i.e., such that

1 + 1 + ... + 1 (p times) = 0
Prove that p is prime.
(b) Prove that every nonzero element of F has order p as an element of the additive group F.
(c) Prove that the number of elements of F is pn for some positive integer n.
(Remark: it can be proved that for each prime p and positive integer n, there is exactly one field with pn elements.)