A note on finite groups with an automorphism inverting or squaring a non-negligible fraction of elements

TL;DR

This paper establishes explicit bounds on the structure of finite groups based on the proportion of elements inverted or squared by an automorphism, linking these fractions to the group's commuting probability and the Fitting subgroup's index.

Contribution

It provides new explicit bounds connecting automorphism element fractions with group structure, improving previous results.

Findings

Bounds on the commuting probability based on automorphism action

Upper bounds on the index of the Fitting subgroup given automorphism conditions

Enhanced structural understanding of finite groups with automorphisms inverting or squaring many elements

Abstract

We show that for a finite group $G$, the commuting probability of $G$ can be explicitly bounded from below in a nontrivial way by a function in the maximum fraction of elements inverted resp. squared by an automorphism of $G$. Using these bounds together with a result of Guralnick and Robinson gives upper bounds on the index of the Fitting subgroup of $G$ under each of the two conditions that $G$ have an automorphism inverting resp. squaring at least $\rho|G|$ many elements in $G$, for $\rho\in\left(0,1\right]$ fixed. This is an improvement on previous results of the author.