Finite Groups

The following sections are included:

• Preface

• Contents

This book is primarily about finite groups. Much of what it covers extends to various classes of infinite groups. Where no extra work is involved we present the results for groups in general, but elsewhere, by restricting ourselves to finite groups, we are able to cover much more ground…

In this chapter groups may be infinite. There after we shall confine our attention to finite groups…

From now on we consider only finite groups. This book revolves around conjugacy. The basic conjugacy theorems on which, in many respects, the rest depend, are Sylow’s (L. Sylow 1832-1918). Sylow, a Norwegian, apparently knew of these results years before he disclosed them. He did this, I understand, upon meeting the French mathematican C. Jordan in Oslo. Jordan immediately realised that these results completely revolutionised group theory, rendering redundant much earlier work, including some of his own. Jordan’s holiday in Oslo was presumably less than relaxing. I expect readers to have seen Sylow’s theorems before, but since they are so crucial I present them in full.

By a Sylow p-subgroup of a finite group G for some prime p, we mean a maximal p-subgroup of G. (This is not always the most useful definition, when dealing with potentially infinite groups, but we are excluding that case here.)

Let G be a group and N a normal subgroup of G. A complement of N in G is a subgroup H of G satisfying H ∩ N = (1) and G = HN (= NH since N is normal in G). We also say that N is a normal complement of H. There are a number of terminologies and notations used for this, often emphasizing different aspects of basically the same group theoretic situation: for example, one can say that G is the split extension of N by H or G is the semi-direct product of N by H and write G = N]H or G = H[N or G = N ⋊ H etc…

So far only Chap. 2 has given us any opportunity for obtaining information about simple groups. In Chaps. 3 and 4 our hypotheses ensured that either the group in question had proper normal subgroups and hence was not simple, or that the conclusions were essentially trivial for simple groups. This present chapter will change this…

So far only Chap. 2 has given us any opportunity for obtaining information about simple groups. In Chaps. 3 and 4 our hypotheses ensured that either the group in question had proper normal subgroups and hence was not simple, or that the conclusions were essentially trivial for simple groups. This present chapter will change this…

In this chapter we briefly stray away from our main theme of conjugacy. In Chap. 5 we saw how information about simple groups in particular can be derived from the fusion pattern of a Sylow subgroup. Our aim here is to indicate how such information can be put together to extract knowledge about finite groups in general…

In this chapter we show how much of Chap. 4 can be generalized. We have obtained too many types of conjugate subgroups to see clearly what is happening. We need a more systematic way of constructing such subgroups. It turns out that there are two basic, and dual, theories at work and we have been seeing snippets here and there of one or the other. The main thrust in this generalization and systemization was the work of W. Gaschutz…

The following sections are included:

• Appendix — Some Proofs for Chapter 0

• Notation

• Bibliography

• Index