Capable groups of prime exponent and class 2, II

TL;DR

This paper investigates the capability of certain p-groups of class two and odd prime exponent, providing new results including a complete classification for 4-generator groups and a rank-based sufficient condition.

Contribution

It introduces new linear algebra and counting techniques to analyze the capability of p-groups, settling the 4-generator case and establishing a rank-based criterion.

Findings

Complete classification of 4-generator groups of this type

A new sufficient condition based on ranks of G/Z(G) and [G,G]

Use of linear algebra and counting arguments in group capability analysis

Abstract

We consider the capability of $p$ groups of class two and odd prime exponent. We use linear algebra and counting arguments to establish a number of new results. In particular, we settle the 4-generator case, and prove a sufficient condition based on the ranks of $G/Z(G)$ and $[G,G]$.