Rings and covered groups.
Let \(G\) be an additive group (not necessarily Abelian) with identity \(0\). The set \(M_0(G)\) of all \(0\)-preserving functions from \(G\) to itself forms a near-ring under pointwise addition and composition of functions. If \(R\subseteq M_0(G)\) is a ring under these operations, then \(R\) is said to be a `subring' of \(M_0(G)\). If \(\mathbf C =\{A_\alpha\}\) is a cover of \(G\) by Abelian subgroups, then \(\mathcal R(C)=\{\rho\in M_0(G):\rho|_{A_\alpha}\in\text{End}(A_\alpha)\) for all \(\alpha\}\) is a subring of \(M_0(G)\), called the `ring determined by the cover \(C\)'. On the other hand, if \(R\) is a subring of \(M_0(G)\), then \(\mathcal C(R)=\{B\leq G:B\) is an Abelian subgroup of \(G\) and \(R|_B\subseteq\text{End}(B)\}\) is an Abelian cover of \(G\), called the `cover of \(G\) determined by \(R\)'. The following theorem is proved: Let \(\Gamma\) denote the collection of Abelian covers of \(G\), and let \(\Lambda\) denote the collection of subrings of \(M_0(G)\). Then the maps \(\mathcal R\colon \Gamma\to\Lambda\); \(C\mapsto\mathcal R(C)\) and \(\mathcal C\colon\Lambda\to\Gamma\); \(R\mapsto\mathcal C(R)\) determine a Galois correspondence between \(\Gamma\) and \(\Lambda\). Further results specify conditions on a cover \(C\) for which \(\mathcal R(C)\) is maximal or simple as a subring of \(M_0(G)\). For example, if \(C\) is a partition of \(G\), or if \(G\) is nilpotent and \(C\) is a cover by cyclic subgroups, then \(\mathcal R(C)\) is a maximal subring of \(M_0(G)\). Furthermore, if \(C\) is a cover of \(G\) with \(|G|=p^4\), \(\exp(G)=p\) (\(p\) a prime) and \(G\notin C\), then \(\mathcal R(C)\) is simple if and only if \(\mathcal R(C)\cong\mathbb{Z}_p\). Finally, results regarding \(\langle I\rangle\), the subring of \(M_0(G)\) generated by the identity map \(I\), are discussed. For example, if \(G\) is nonabelian, \(|G|=p^3\), \(p\) an odd prime, and \(A\) is the cover of \(G\) by Abelian subgroups, then \(\mathcal R(A)=\langle I\rangle\) if and only if \(\exp(G)=\exp(Z(G))\). Also, if \(A\) is the cover of \(D_n\) (the dihedral group of order \(2n\)) by Abelian subgroups, then \(\mathcal R(A)=\langle I\rangle\) if and only if \(n\) is even. -- Several examples are given.
scientific article; zbMATH DE number 5353586
scientific article