On $p$-groups with automorphism groups related to the Chevalley group $G_2(p)$

TL;DR

This paper constructs the smallest known p-group of nilpotency class two with automorphism group related to G_2(p), analyzing its properties via octonion algebra actions and module reducibility.

Contribution

It introduces a minimal p-group with specific automorphism properties linked to G_2(p), including explicit construction and classification for p=3.

Findings

Constructed the smallest p-group with the given automorphism properties.

Analyzed the action of G_2(q) on octonion algebra over finite fields.

Identified two nonisomorphic groups when p=3.

Abstract

Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\mathrm{Aut}(P)$ induces $N_{\mathrm{GL}(7,p)}(G_2(p)) = Z(\mathrm{GL}(7,p)) G_2(p)$ on the Frattini quotient $P/\Phi(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p = 3$, our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_2(q)$ on the octonion algebra over $\mathbb{F}_q$, for each power $q$ of $p$, and explore the reducibility of the exterior square of each irreducible seven-dimensional $\mathbb{F}_q[G_2(q)]$-module.