Semi-extraspecial groups with an abelian subgroup of maximal possible order

TL;DR

This paper studies semi-extraspecial and ultraspecial p-groups, providing a framework for their construction, and generalizes semifields to classify certain semi-extraspecial groups with abelian subgroups.

Contribution

It proves that all p-groups of nilpotence class 2 embed into ultraspecial groups and introduces a generalized semifield concept for classifying semi-extraspecial groups.

Findings

Every p-group of nilpotence class 2 embeds into an ultraspecial group.

Constructs all ultraspecial groups of order p^{3n} with a large abelian subgroup.

Generalizes semifields to classify semi-extraspecial groups as products of abelian subgroups.

Abstract

Let $p$ be a prime. A $p$-group $G$ is defined to be semi-extraspecial if for every maximal subgroup $N$ in $Z(G)$ the quotient $G/N$ is a an extraspecial group. In addition, we say that $G$ is ultraspecial if $G$ is semi-extraspecial and $|G:G'| = |G'|^2$. In this paper, we prove that every $p$-group of nilpotence class $2$ is isomorphic to a subgroup of some ultraspecial group. Given a prime $p$ and a positive integer $n$, we provide a framework to construct of all the ultraspecial groups order $p^{3n}$ that contain an abelian subgroup of order $p^{2n}$. In the literature, it has been proved that every ultraspecial group $G$ order $p^{3n}$ with at least two abelian subgroups of order $p^{2n}$ can be associated to a semifield. We provide a generalization of semifield, and then we show that every semi-extraspecial group $G$ that is the product of two abelian subgroups can be associated with this generalization of semifield.