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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. | Turk J Math (2016) 40: 1340 – 1348 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1504-57 Research Article Rings associated to coverings of finite p-groups 1 Gary WALLS1,∗, Linhong WANG2,∗ Department of Mathematics, Southeastern Louisiana University, Hammond, LA, USA 2 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA Received: 20.04.2015 • Accepted/Published Online: 19.02.2016 • Final Version: 02.12.2016 Abstract: 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. Key words: Finite p -groups, covers of groups, rings of functions 1. Introduction Covers of groups by subgroups and rings of functions that act as endomorphisms on each subgroup were studied in many papers including [1, 2, 4, 5]. Definition 1.1 Suppose that G is a group and C is a collection of subgroups of G . We say that C is a cover ∪ of G provided C∈C C = G . If all the elements of C have a certain property γ , we say that C is a γ -covering of G . It is well known, e.g., [3], that the endomorphisms of a nonabelian group G do not necessarily form a ring under the operations of function addition and composition. Coverings by abelian subgroups are used to obtain rings of functions on G. Definition .