Elements with bounded centralizer order. Let G be a finite group containing a subgroup A of order ≤ n with centralizer CG(A) of order ≤ k. What can be said about G in terms of these parameters, possibly also involving the structure of A? In this paper we describe, without proofs, some recent results on this question. Some of them, not surprisingly, have applications to locally finite groups. A slightly different question, with a similar flavour, is discussed in §2.

The most famous theorem of this type is the classical one of Brauer and Fowler, which asserts that if G is a finite simple group containing an involution i, then ∣G∣ is bounded in terms of ∣CG(i)∣. More generally, similar arguments show that the same holds if i is an involutory automorphism of G and CG(i) is the fixed point group of i, provided of course that G is nonabelian. The proofs of these facts are elegant and essentially elementary. We give below the generalization to automorphisms of arbitrary order. This uses the classification of finite simple groups, which we have no compunction in using where convenient in this paper.

Theorems that bound some invariant of finite groups in terms of others often have consequences in locally finite group theory. For example, it is not too difficult to deduce from the automorphism form of the Brauer-Fowler Theorem that a locally finite group containing an involution with finite centralizer has a locally soluble subgroup of finite index, and much stronger results are true (see below). The kind of results we look for are selected with such applications in mind.