Semi-extraspecial Groups

TL;DR

This paper surveys semi-extraspecial p-groups, exploring their properties, connections to other group classes, and bounds on their structure, while introducing ultraspecial groups and their relation to semifields.

Contribution

It provides a comprehensive overview of semi-extraspecial groups, establishes bounds on their structure, and links ultraspecial groups to semifields, expanding understanding of these algebraic objects.

Findings

Connections between semi-extraspecial groups and Camina groups, VZ-groups.

Upper bounds on the order of centers and abelian normal subgroups.

Ultraspecial groups relate to semifields when certain conditions are met.

Abstract

We survey the results regarding semi-extraspecial $p$-groups. Semi-extraspecial groups can be viewed as generalizations of extraspecial groups. We present the connections between semi-extraspecial groups and Camina groups and VZ-groups, and give upper bounds on the order of the center and the orders of abelian normal subgroups. We define ultraspecial groups to be semi-extraspecial groups where the center is as large as possible, and demonstrate a connection between ultraspecial groups that have at least two abelian subgroups whose order is the maximum and semifields.