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

TL;DR

This paper investigates finite groups with automorphisms that invert, square, or cube a significant fraction of elements, showing such groups are structurally close to abelian or solvable depending on the automorphism type.

Contribution

It establishes bounds on the structure of finite groups based on the proportion of elements affected by automorphisms inverting, squaring, or cubing, extending the understanding of group near-abelianity.

Findings

Groups with automorphisms inverting or squaring at least ρ|G| elements are almost abelian.

Groups with automorphisms cubing at least ρ|G| elements are almost solvable.

Bounds on the index and derived length of the solvable radical are provided.

Abstract

There are various results in the literature which are part of the general philosophy that a finite group for which a certain parameter (for example, the number of conjugacy classes or the maximum number of elements inverted, squared or cubed by a single automorphism) is large enough must be close to being abelian. In this paper, we show the following: Fix a real number $\rho$ with $0<\rho\leq 1$. Then a finite group $G$ with an automorphism inverting or squaring at least $\rho|G|$ of the elements in $G$ is "almost abelian" in the sense that both the index and the derived length of the solvable radical of $G$ are bounded. Furthermore, if $G$ has an automorphism cubing at least $\rho|G|$ of the elements in $G$, then $G$ is "almost solvable" in the sense that the index of the solvable radical of $G$ is bounded.