Rings associated to coverings of finite p-groups

TL;DR

This paper investigates the structure of a specific ring of functions on finite p-groups, constructed via covers by cyclic or elementary abelian subgroups, and provides an algorithm to determine its structure.

Contribution

It introduces an algorithm to determine the ring of functions arising from covers of finite p-groups by specific abelian subgroups and describes its structure as a subdirect product.

Findings

Algorithm for identifying elements of the function ring

Structure of the ring as a subdirect product

Graph construction linking subgroups and covers

Abstract

In general the endomorphisms of a non-abelian 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 which allows us to determine the elements of the ring of functions of a finite $p$-group which 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.