On the commuting probability for subgroups of a finite group

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