1 results
ON
$p$-GROUPS WITH AUTOMORPHISM GROUPS RELATED TO THE CHEVALLEY GROUP 
$G_{2}(p)$
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 108 / Issue 3 / June 2020
- Published online by Cambridge University Press:
- 08 January 2020, pp. 321-331
- Print publication:
- June 2020
-
- Article
- Export citation
-
Let
$p$ be an odd prime. We construct a 
$p$-group 
$P$ of nilpotency class two, rank seven and exponent 
$p$, such that 
$\text{Aut}(P)$ induces 
$N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$ on the Frattini quotient 
$P/\unicode[STIX]{x1D6F7}(P)$. 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.
