On finite groups where the order of every automorphism is a cycle length

TL;DR

This paper investigates finite groups where each automorphism has a cycle length equal to its order, providing new conditions for this property and exploring its presence in various groups.

Contribution

It offers a new proof of a known result for nilpotent groups, introduces two sufficient conditions for the property, and analyzes the minimal group order violating it.

Findings

Automorphisms of finite nilpotent groups have cycles matching their order.

Groups of order less than 120 always satisfy the property.

Any finite group can embed into groups with and without this property.

Abstract

Using Frobenius normal forms of matrices over finite fields as well as the Burnside Basis Theorem, we give a direct proof of Horo\v{s}evski\u{i}'s result that every automorphism $\alpha$ of a finite nilpotent group has a cycle whose length coincides with $\mathrm{ord}(\alpha)$. Also, we give two new sufficient conditions for an automorphism $\alpha$ of an arbitrary finite group to satisfy this property, namely when $\mathrm{ord}(\alpha)$ is a product of at most two prime powers or when $\alpha$ has a sufficiently large cycle. This will allow us to show that the least order of a group where this property is violated is 120. Finally, we observe that any finite group embeds both into a group with this property (as all finite symmetric groups enjoy the property) as well as into a finite group not having this property.