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

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.

arXiv comment10 pages, 1 figure

Cited by in corpus (1)

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