Finite groups with an automorphism of large order

TL;DR

This paper investigates finite groups with automorphisms of large order, establishing conditions under which the group is abelian or solvable, and providing bounds on the group's structure based on automorphism properties.

Contribution

The paper generalizes previous results on automorphism cycle length, proving new bounds and structural implications for finite groups with automorphisms of large order.

Findings

If automorphism order > 1/2 of group size, then the group is abelian.

If automorphism order > 1/10 of group size, then the group is solvable.

The ratio of maximum automorphism order to maximum cycle length can be arbitrarily large.

Abstract

Let $G$ be a finite group, and assume that $G$ has an automorphism of order at least $\rho|G|$, with $\rho\in\left(0,1\right)$. Generalizing recent analogous results of the author on finite groups with a large automorphism cycle length, we prove that if $\rho>1/2$, then $G$ is abelian, and if $\rho>1/10$, then $G$ is solvable, whereas in general, the assumption implies $[G:\operatorname{Rad}(G)]\leq\rho^{-1.78}$, where $\operatorname{Rad}(G)$ denotes the solvable radical of $G$. Furthermore, we generalize an example of Horo\v{s}evski\u{\i} to show that in finite groups, the quotient of the maximum automorphism order by the maximum automorphism cycle length may be arbitrarily large.