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

TL;DR

This paper characterizes certain semi-extraspecial p-groups with large automorphism order, showing they are isomorphic to Sylow p-subgroups of special unitary or special linear groups, revealing their algebraic structure.

Contribution

It establishes a classification of semi-extraspecial p-groups with automorphisms of large order as Sylow p-subgroups of specific classical groups.

Findings

Groups are isomorphic to Sylow p-subgroups of SU_3(p^{2a}) or SL_3(p^a).

Automorphism of order |G:G'|-1 characterizes these groups.

Provides a structural link between p-groups and classical groups.

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 ${\rm SL}_3 (p^a)$.