  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.
