Kernels of covered groups (Q1078300)

Consider an additively expressed group \(G\) with identity element 0. A collection \(\mathcal C\) of subgroups of \(G\) is a cover for \(G\) if (1) \(\cup\, \mathcal C=G\) and (2) for \(H,K\in\mathcal C\), \(H\ne \{0\}\) or \(G\), if \(K\ne H\), then \(H\not\subseteq K\). The pair \((G,\mathcal C)\) is called a covered group. If (2) is replaced by (3), for each \(H,K\in\mathcal C\) either \(H=K\) or \(H\cap K=\{0\}\), then \(\mathcal C\) is a fibration and \((G,\mathcal C)\) is a fibered group. The near-ring d.g. \(E(G,\mathcal C)=\{f\in \text{End}(G)\mid f(H)\subseteq H\) for each \(H\in\mathcal C\}\) is called the kernel of \((G, \mathcal C)\). This article is a continuation of the authors' study on relating the algebraic properties of the kernel with the geometric structure derived from \((G,\mathcal C)\). In previous investigations, attention was restricted to covers that were fibrations. That restriction is lifted here. The objective is to obtain structural information about d.g. \(E(G,\mathcal C)\) from properties of \((G,\mathcal C)\). Attention is focused on finite abelian groups (in particular elementary abelian \(p\)-groups) and the questions on whether or not \(E(G,\mathcal C)\) is a semisimple ring, a local ring, or a field. After establishing several general properties, conditions become more restrictive. The examples are excellent and relate to interesting questions on rings of matrices over a field of \(p\) elements.