To my mind the theorem classifying the finite simple groups is the most important result in finite group theory. As I indicated in the preface, the Classification Theorem is the foundation for a powerful theory of finite groups which proceeds by reducing suitable group theoretical questions to questions about representations of simple groups. The final chapter of this book is devoted primarily to a discussion of the Classification Theorem and the finite simple groups themselves.

Sections 45 and 46 introduce two classes of techniques useful in the study of simple groups. Section 45 investigates consequences of the fact that each pair of involutions generates a dihedral group. The two principal results of the section are the Thompson Order Formula and the Brauer–Fowler Theorem. The Thompson Order Formula supplies a formula for the order of a finite group with at least two conjugacy classes of involutions in terms of the fusion of those involutions in the centralizers of involutions. The Brauer-Fowler Theorem shows that there are at most a finite number of finite simple groups possessing an involution whose centralizer is isomorphic to any given group.

Section 46 considers the commuting graph on the set of elementary abelian p-subgroups of p-rank at least k in a group G. The determination of the groups for which this graph is disconnected for small k plays a crucial role in the Classification Theorem.

Chapter 8 investigates p-groups from two points of view: first through a study of p-groups which are extremal with respect to one of several parameters (usually connected with p-rank) and second through a study of the automorphism group of the p-group.

Recall that if p is a prime then the p-rank of a finite group is the maximum dimension of an elementary abelian p-subgroup, regarded as a vector space over GF(p). Section 23 determines p-groups of p-rank 1, p-groups in which each normal abelian subgroup is cyclic, and, for p odd, p-groups in which each normal abelian subgroup is of p-rank at most 2. Perhaps most important, the p-groups of symplectic type are determined (a p-group is of symplectic type if each of its characteristic abelian subgroups is cyclic).

The Frattini subgroup is introduced to study p-groups and their automorphisms. Most attention is focused on p′-groups of automorphisms of p-groups; a variety of results on the action of p′-groups on p-groups appear in section 24. One very useful result is the Thompson A × B Lemma. Also of importance is the concept of a critical subgroup.

Extremal p-groups

In this section p is a prime and G is a p-group.

The Frattini subgroup of a group H is defined to be the intersection of all maximal subgroups of H. Φ(H) denotes the Frattini subgroup of H.

If G is a finite group, H ≤ G, and α: H → A is a homomorphism of H into an abelian group A, then it is possible to construct a homomorphism V : G → A from α in a canonical way. V is called the transfer of G into A via α. If we can show there exists g ∈ G – ker(V), then, as G/ker(V) is abelian, g ∉ G(1) the commutator group of G. In particular G is not nonabelian simple.

It is however in general difficult to calculate gV explicitly and decide whether g ∈ ker(V). To do so we need information about the fusion of g in H; that is information about gG∩H. Hence chapter 13 investigates both the transfer map and techniques for determining the fusion of elements in subgroups of G.

Section 38 contains a proof of Alperin's Fusion Theorem, which says that p-local subgroups control the fusion of p-elements. To be somewhat more precise, if P is a Sylow p-subgroup of G then we can determine when subsets of P are fused in G (i.e. conjugate in G) by inspecting the p-locals H of G with P ∩ H Sylow in H.

Section 39 investigates normal p-complements. A normal p-complement for a finite group G is a normal Hall p′-subgroup of G. Various criteria for the existence of such objects are generated, The most powerful is the Thompson Normal p-Complement Theorem, which is used in the next section to establish the nilpotence of Frobenius kernels.