On the capability of finite groups of class two and prime exponent

TL;DR

This paper investigates the capability of finite p-groups of class two and odd prime exponent, translating the problem into linear algebra, and provides new criteria and classifications for capability within this group class.

Contribution

It introduces new necessary and sufficient conditions for capability, including rank-based criteria, and characterizes capable groups among 5-generated groups of this class.

Findings

Established new criteria for group capability.

Proved capability conditions depend on ranks of specific subgroups.

Characterized capable groups among 5-generated groups.

Abstract

We consider the capability of $p$-groups of class two and odd prime exponent. The question of capability is shown to be equivalent to a statement about vector spaces and linear transformations, and using the equivalence we give proofs of some old results and several new ones. In particular, we establish a number of new necessary and new sufficient conditions for capability, including a sufficient condition based only on the ranks of $G/Z(G)$ and $[G,G]$. Finally, we characterise the capable groups among the 5-generated groups in this class.