Research ArticleNo Access

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

Alexander Bors

Mathematics Department, University of Salzburg, Hellbrunner Straße 34, 5020 Salzburg, Austria

E-mail Address: alexander.bors@sbg.ac.at

https://doi.org/10.1142/S0219498819500555Cited by:1 (Source: Crossref)

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

Communicated by R. Guralnick

Keywords:

AMSC: 20D25, 20D45, 20D60, 05D05, 12E20, 20C15, 20D05

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