Finite groups with an automorphism of large order

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.