A bound on element orders in the holomorph of a finite group

TL;DR

This paper establishes an upper bound on the orders of elements in the holomorph of a finite group, showing they do not exceed the group's order, with implications for automorphism order bounds.

Contribution

It provides a new theoretical bound on element orders in the holomorph of finite groups, linking it to the group's order and automorphism properties.

Findings

Element orders in Hol(G) are bounded by |G|

Application to automorphism order bounds

Theoretical proof of the bound

Abstract

Let $G$ be a finite group. We prove a theorem implying that the orders of elements of the holomorph $\operatorname{Hol}(G)$ are bounded from above by $|G|$, and we discuss an application to bounding automorphism orders of finite groups.