On the commuting probability for subgroups of a finite group

The probability that two randomly chosen elements of a finite group $G$ commute is denoted by $Pr(G)$. If $K\leq G$, we denote by $Pr(K,G)$ the probability that an element of $G$ commutes with an element of $K$. A well known theorem, due to P. M. Neumann, says that if $G$ is a finite group such that $Pr(G)\geq\epsilon$, then $G$ has a nilpotent normal subgroup $T$ of class at most $2$ such that both the index $[G:T]$ and the order $|[T,T]|$ of the commutator subgroup of $T$ are $\epsilon$-bounded. The main purpose of this paper is to establish the following stronger forms of P. M. Neumann's theorem. 1. Let $\epsilon>0$, and let $K$ be a subgroup of a finite group $G$ such that $Pr(K,G)\geq\epsilon$. Let $H=\langle K^G\rangle$ be the normal closure of $K$ in $G$. Then $G$ has a nilpotent normal subgroup $T\leq H$ of nilpotency class at most $2$ such that both the index $[H:T]$ and the order $|[T,T]|$ of the commutator subgroup are $\epsilon$-bounded. 2. Let $\epsilon>0$, and let $H$ be a normal subgroup of a finite group $G$ containing $\gamma_k(G)$ for some $k\geq1$. Suppose that $Pr(H,G)\geq\epsilon$. Then $G$ has a nilpotent normal subgroup $T$ of nilpotency class at most $k+1$ such that both the index $[G:T]$ and the order of $\gamma_{k+1}(T)$ are $\epsilon$-bounded. We also deduce a number of corollaries of these results.