On the commuting probability for subgroups of a finite group

TL;DR

This paper investigates the probability that elements of a finite group commute with elements of a subgroup, establishing bounds on related subgroup structures when this probability exceeds a fixed threshold.

Contribution

It generalizes Neumann's theorem by providing bounds on subgroup indices and commutator subgroup orders based on commuting probability for subgroups.

Findings

Existence of a normal subgroup with bounded index related to the commuting probability.

Bounded order of the commutator subgroup in the constructed subgroup.

Applicability to various subgroup types like the Fitting subgroup and Sylow subgroups.

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 indexes $[G:T]$ 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.