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.
Finite $p$-groups, covers of groups, rings of functions
WALLS, GARY and WANG, LINHONG
"Rings associated to coverings of finite $p$-groups,"
Turkish Journal of Mathematics: Vol. 40:
6, Article 14.
Available at: https://journals.tubitak.gov.tr/math/vol40/iss6/14