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.