Mathematics 893 Problem Set 4

Suggested reading: Sections 3.2, 3.3, 4.1, 4.2, 4.3, 4.5 (except for omissions noted below). Optional extra thinking: Proposition 1, page 114. The proof of this result is scattered through section 1.7 and pages 112-114. This is a more sophisticated way of thinking about group actions, which I have avoided. However sometimes the action of a group G on a set (or mathematical structure of some kind) A is defined as a homomorphism of G into the automorphisms of A, so it would be a good idea for you to be aware of this approach and its relation with the definition we used (that on page 41).

Questions due next Friday (October 14):

1. Page 95 # 8, 10, 12, 20

2. Page 101 #7

3. Page 111 #4,5 Do (1,3) and (1,2,3,4) generate S_{4}? Explain.

4. Page 117, # 9

5. Page 146 # 8, 18, 32

6. Bonus problem. Let G be an infinite group and H a subgroup of finite index in G. Prove that G has a normal subgroup of finite index which is contained in H.

Notes: Our immediate goal is to prove the Sylow Theorems (section 4.5). A few topics that we do not need for this will be omitted, for example the Third and Fourth Isomorphism Theorems (in section 3.3) and most of sections 3.4 and 3.5. We do need Proposition 21 from section 3.4 (page 102). For now we will assume this result. It will be proved by a different method when we discuss abelian groups as modules over the ring of integers. Some of the omitted topics may be discussed later, either because we need them, because they can be done more easily when we have more tools to work with, or for comparison with the corresponding properties of rings.