On the commuting probability for subgroups of a finite group

Abstract

Let$K$be a subgroup of a finite group$G$. The probability that an element of$G$commutes with an element of$K$is denoted by$Pr(K,G)$. Assume that$Pr(K,G)\geq \epsilon$for some fixed$\epsilon >0$. We show that there is a normal subgroup$T\leq G$and a subgroup$B\leq K$such that the indices$[G:T]$and$[K:B]$and the order of the commutator subgroup$[T,B]$are$\epsilon$-bounded. This extends the well-known theorem, due to P. M. Neumann, that covers the case where$K=G$. We deduce a number of corollaries of this result. A typical application is that if$K$is the generalized Fitting subgroup$F^{*}(G)$then$G$has a class-2-nilpotent normal subgroup$R$such that both the index$[G:R]$and the order of the commutator subgroup$[R,R]$are$\epsilon$-bounded. In the same spirit we consider the cases where$K$is a term of the lower central series of$G$, or a Sylow subgroup, etc.