In this introductory chapter we give an overview of the strategy behind the
proof of the theorem classifying the finite simple groups, while for the most part
attempting to avoid the fine structure of the proof. At various places we make
oversimplifications to avoid technicalities; this means that some statements are not
quite true in special cases, but hopefully this approach conveys the flavor of the
proof more clearly than a more involved, technically correct discussion.
To be a bit more specific, we describe how, for a suitable choice of prime p, each
finite simple group G is determined up to isomorphism by its p-local subgroups.
This makes it possible to classify the simple groups in terms of their local structure.
We explain why the prime 2 plays a special role and is usually the optimal
choice for p. We sketch a proof of the Dichotomy Theorem 0.3.10, which partitions
the simple groups into groups of Gorenstein-Walter type and characteristic 2 type,
according to two possible 2-local structures. We also describe the partition of each
of these two types of simple groups into large and small groups of the given type,
leading to a four-part subdivision of the proof of the Classification.
0.1. The Classification Theorem
We begin with a statement of the Classification Theorem:
Theorem 0.1.1 (Classification Theorem). Each finite simple group is isomor-
phic to one of:
(1) A group of prime order.
(2) An alternating group on a set of order at least 5.
(3) A finite simple group of Lie type.
(4) One of 26 sporadic simple groups.
In Section 0.4 we discuss each of the four classes of groups in more detail. But first
we begin with a brief discussion of local group theory, and the role it plays in the
proof of the Classification. We assume the reader is familiar with the most basic
notation and terminology from finite group theory. For those who are not, some
notation and terminology is defined at the start of Section A.1 in Appendix A.
Throughout this chapter, G is a finite group and p is a prime. Further Sylp(G)
denotes the set of all Sylow p-subgroups of G. Pick S ∈ Sylp(G), and write P for
the set of nontrivial subgroups of S.
A subgroup of G is called p-local if it is the normalizer NG(P ) of some nontrivial
p-subgroup P of G. We speak simply of a local subgroup if we do not wish to
specify the prime p. The study of finite groups from the point of view of their local
subgroups is called local group theory.