Covering groups of rank 1 of elementary abelian groups

Covering groups of elementary Abelian groups of odd exponent p can be classified according to the rank of their p th power homomorphisms, which may be regarded as linear transformations of p -vector spaces. This article contains a description of the isomorphism types and the automorphism groups of those covering groups in which this rank is 1. Analogous considerations of elementary Abelian 2-groups and their covering groups are included in the final section.