Finite groups with an automorphism inverting, squaring or cubing a non-negligible fraction of elements

Abstract

Finite groups with an automorphism mapping a sufficiently large proportion of elements to their inverses, squares and cubes have been studied for a long time, and the gist of the results on them is that they are “close to being abelian”. In this paper, we consider finite groups [Formula: see text] such that, for a fixed but arbitrary [Formula: see text], some automorphism of [Formula: see text] maps at least [Formula: see text] many elements of [Formula: see text] to their inverses, squares and cubes. We will relate these conditions to some parameters that measure, intuitively speaking, how far the group [Formula: see text] is from being solvable, nilpotent or abelian; most prominently the commuting probability of [Formula: see text], i.e. the probability that two independently uniformly randomly chosen elements of [Formula: see text] commute. To this end, we will use various counting arguments, the classification of the finite simple groups and some of its consequences, as well as a classical result from character theory.