On the commuting probability of p-elements in a finite group

Abstract

Let G be a finite group, let p be a prime and let Pr p (G) be the probability that two random p-elements of G commute.In this paper we prove that Pr p (G) > ( p 2 + p -1)/ p 3 if and only if G has a normal and abelian Sylow p-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group.This bound is best possible in the sense that for each prime p there are groups with Pr p (G) = ( p 2 + p -1)/ p 3 and we classify all such groups.Our proof is based on bounding the proportion of p-elements in G that commute with a fixed p-element in G \ O p (G), which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.

Locations

• Algebra & Number Theory - View - PDF
• arXiv (Cornell University) - View - PDF