On the commuting probability for subgroups of a finite group
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.