On $p$-groups with automorphism groups related to the exceptional Chevalley groups

TL;DR

This paper investigates the structure of certain $p$-groups related to exceptional Chevalley groups by analyzing their automorphism groups and module structures, leading to the construction of minimal $p$-groups with specific automorphism properties.

Contribution

It introduces a method to construct minimal $p$-groups whose automorphism groups relate to exceptional Chevalley groups, expanding understanding of their overgroup and module structures.

Findings

Constructed minimal $p$-groups with automorphism groups related to Chevalley groups.

Analyzed the submodule structure of exterior squares and Lie powers of modules.

Identified conditions under which automorphism groups match Chevalley groups or their normalizers.

Abstract

Let $\hat G$ be the finite simply connected version of an exceptional Chevalley group, and let $V$ be a nontrivial irreducible module, of minimal dimension, for $\hat G$ over its field of definition. We explore the overgroup structure of $\hat G$ in $\mathrm{GL}(V)$, and the submodule structure of the exterior square (and sometimes the third Lie power) of $V$. When $\hat G$ is defined over a field of odd prime order $p$, this allows us to construct the smallest (with respect to certain properties) $p$-groups $P$ such that the group induced by $\mathrm{Aut}(P)$ on $P/\Phi(P)$ is either $\hat G$ or its normaliser in $\mathrm{GL}(V)$.