Bounding the exponent of a finite group by the exponent of the automorphism group and a theorem of Schur

TL;DR

This paper establishes bounds on the exponent of finite p-groups based on the exponent of their automorphism groups, extending Schur's theorem and providing specific bounds for groups of certain classes.

Contribution

It introduces new bounds relating the exponent of a finite p-group to that of its automorphism group, generalizing classical results like Schur's theorem.

Findings

For class c p-groups, exponent divides p^{ceil(log_p c)} q^3.

For metabelian p-groups of class at most 2p-1, exponent divides pq^3.

Provides bounds linking automorphism group properties to the structure of p-groups.

Abstract

Assume $G$ is a finite $p$-group, and let $S$ be a Sylow $p$-subgroup of $\operatorname{Aut}(G)$ with $\exp(S)=q$. We prove that if $G$ is of class $c$, then $\exp(G)|p^{\ceil{\log_pc}}q^3$, and if $G$ is a metabelian $p$-group of class at most $2p-1$, then $\exp(G)|pq^3$.