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

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.