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

摘要

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 G such that, for a fixed but arbitrary rho is an element of (0, 1], some automorphism of G maps at least rho vertical bar G vertical bar many elements of G to their inverses, squares and cubes. We will relate these conditions to some parameters that measure, intuitively speaking, how far the group C is from being solvable, nilpotent or abelian; most prominently the commuting probability of G, i.e. the probability that two independently uniformly randomly chosen elements of G 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.