2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/216253We 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.43 pp; incorporates results from older paper; fix amsrefs/hyperref incompatibility and a typoGroup Theory20D15, 20F12text