Groups with bounded centralizer chains and the Borovik–Khukhro conjecture

Abstract

Abstract Let G be a locally finite group and let F ⁢ ( G ) {F(G)} be the Hirsch–Plotkin radical of G. Let S denote the full inverse image of the generalized Fitting subgroup of G / F ⁢ ( G ) {G/F(G)} in G. Assume that there is a number k such that the length of every nested chain of centralizers in G does not exceed k. The Borovik–Khukhro conjecture states, in particular, that under this assumption, the quotient G / S {G/S} contains an abelian subgroup of finite index bounded in terms of k. We disprove this statement and prove a weak analogue of it.