Suppose G is a p-group which is not necessarily finite but has non- trivial centre Z. The paper is concerned with the case when G has a partition, that is, a set of at least two subgroups for which any non- identity element of G lies in precisely one of them. The case of exponent p is not of much interest, since the set of all subgroups of order p is such a partition, but it is easy to see that if the exponent of G is greater than p, the subgroup \(H_ p\) generated by all elements of order greater than p is contained in a member of the partition. Thus \(H_ p<G\), and conversely, groups for which \(H_ p<G\) have an obvious partition. Consideration of such groups leads to the definition of a group of type (p,\(\nu)\) as \(<a,b>\), when \(| <a>| =p^{\nu}\), ab\(\neq ba\), \(b\not\in <a>^ G\), and \((b^ ix)=1\) for \((i,p)=1\) and \(x\in <a>^ G\). Such groups are described in terms of a class of commutative local rings. In the final section, some examples are constructed by means of couplings. The readability of the paper is marred by its unorthodox notation.