In this chapter we aim to prove the general version of Pyber's theorem: the proof is contained in the final section. The three sections preceding the proof each deal with a different ingredient that is needed there. Section 16.1 contains theorems that bound the number of generators of a group in various contexts. Section 16.2 is concerned with central extensions (especially of perfect groups). Finally, in Section 16.3 we define and explore the notion of the generalised Fitting subgroup of a group.

Three theorems on group generation

This section contains proofs of three theorems, each of which makes statements about the existence of certain kinds of generating sets for finite groups. The first, due to Wolfgang Gaschütz, [35], will be needed to prove the third theorem of this section. The second and third depend on the Classification of Finite Simple Groups; they will be used in the proof of the general case of Pyber's theorem in Section 16.4.

Theorem 16.1Let G be a finite group, and let N be a normal subgroup of G. Suppose that G may be generated by r elements, and let g1, g2 …, gr ∈ G be such that g1N, g2N, …, grN generate G/N. Then there exist generators {h1, h2 …, hr} for G such that hi ∈ giN for i ∈ {1, 2, … r}.