Preliminaries

Abstract

In this chapter we collect together results which will be applied throughout the monograph. We have deliberately included well-known results to aid non-expert readers, though we do assume a degree of familiarity with general group theory. This chapter contains the vast majority of the background results necessary to understand the material up to and including Chapter 8. For the more elementary results and their proofs see either the first six chapters of Gorenstein’s book [61], Aschbacher’s book [5] or Suzuki’s book [177] while [79, 90,178] contain more advanced material particularly relevant to this monograph. We have given references after each of the foundational results; however, we want to point out that although most of the results can be found in all of the above mentioned books, we have just chosen references which coincide with our choice of statement and these are mainly from [61] or [5]