Semi-extraspecial $p$-groups with automorphisms of large order

TL;DR

This work classifies semi-extraspecial $p$-groups that admit a $p'$-automorphism of maximal possible order $|G:G'|-1$. By deriving a tight bound on automorphism orders and analyzing the induced actions on $G/Z(G)$ through bilinear forms, the authors show such $G$ must be ultraspecial and realize exactly as Sylow $p$-subgroups of ${\rm SU}_3(p^{2a})$, which for odd $p$ coincide with Sylow $p$-subgroups of ${\rm SL}_3(p^a)$. The proof combines transitivity properties on cosets, the structure of $Z(G)$, and a Beisiegel-type result to identify $G$ with a Sylow $p$-subgroup of ${\rm SU}_3(p^{2a})$, yielding a complete classification and clarifying the $p=2$ exception. The results connect semi-extraspecial Camina $p$-groups to unitary and special linear groups, with implications for the structure of group algebras and constructions via central products.

Abstract

In this paper, we consider semi-extraspecial $p$-groups $G$ that have an automorphism of order $|G:G'| - 1$. We prove that these groups are isomorphic to Sylow $p$-subgroups of ${\rm SU}_3 (p^{2a})$ for some integer $a$. If $p$ is odd, this is equivalent to saying that $G$ is isomorphic to a Sylow $p$-subgroup of ${\rm SL}_3 (p^a)$.

Theorems & Definitions (21)

• Theorem A
• Corollary B
• Lemma 2.1: BEI77 and brown
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4