Kernels of covered groups. II
[Part I cf. \textit{H. Karzel} and the authors, ibid. 9, 70-81 (1986; Zbl 0596.16031).] A set \(C=\{G_ a|\) \(a\in A\}\), A an index set, of subgroups of a group \(<G,+>\) is a cover of G if \(\cup_{A}G_ a=G\) and each \(G_ a\) is a proper nontrivial subgroup of G. The set \(N(G,C)=\{f\in End(G)|\) \(f(G_ a)\subseteq G_ a\), \(\forall G_ a\in C\}\) is a semigroup with zero, and under the operation of function composition has an identity element. The near-ring dgN(G,C) is distributively generated by N(G,C) is called the kernel of the covered group (G,C). For a near-ring R of mappings of a group G into itself, a sublattice of G is considered, namely, \(Lat(R,G)=\{H\subseteq G|\) f(H)\(\subseteq H\), \(\forall f\in R\}\). In this article, the focus is on elementary abelian p-groups. Consequently \(N(G,C)=dgN(G,C)\) is a ring. Necessary and sufficient conditions are found in terms of Lat(dgN(G,C),G) for the kernel to be a field, or a simple ring, or a semisimle ring. In addition, starting with the group \(G=(Z_ p)^ n\), the authors give constructions for the fields and simple rings which arise from the covers of G. The case that N is a local ring also is discussed. There are an adequate number of examples which bring the concepts into focus, especially those which have a geometric flavor.
scientific article; zbMATH DE number 4129016
scientific article