Rings associated to coverings of finite p-groups

In general the endomorphisms of a nonabelian group do not form a ring under the operations of addition and composition of functions. Several papers have dealt with the ring of functions defined on a group, which are endomorphisms when restricted to the elements of a cover of the group by abelian subgroups. We give an algorithm that allows us to determine the elements of the ring of functions of a finite p-group that arises in this manner when the elements of the cover are required to be either cyclic or elementary abelian of rank 2. This enables us to determine the actual structure of such a ring as a subdirect product. A key part of the argument is the construction of a graph whose vertices are the subgroups of order p and whose edges are determined by the covering.