Rings determined by cyclic covers of groups. (Q405898)

Let \((G,+)\) be a finite additive group, not necessarily Abelian. Let \(\mathcal C=\{A_1,A_2,\ldots,A_n\}\) be a collection of maximal cyclic subgroups of \(G\), such that \(\bigcup_{i=1}^nA_i=G\), i.e., \(\mathcal C\) is a \textit{cover} of \(G\). The aim of the paper is to investigate the structure of the ring \(\mathcal R(\mathcal C)=\{f\colon G\to G\mid f|_{A_i}\in\mathrm{End}(A_i),\;\forall i=1,2,\ldots,n\}\), associated with the cover \(\mathcal C\). The operations of the ring are pointwise addition and composition of functions.NEWLINENEWLINE It is shown that the ring \(\mathcal R(\mathcal C)\) is semisimple if and only if the order of every element of \(G\) is square-free. Furthermore, \(\mathcal R(\mathcal C)\) is simple if and only if \(G\) is a cyclic group of prime order \(p\) if and only if \(\mathcal R(\mathcal C)\cong\mathbb Z_p\) if and only if \(\mathcal R(\mathcal C)\) is a field.NEWLINENEWLINE Results are also given for the particular cases where \(G=S_n\), \(G=A_n\) (the symmetric and alternating groups) and \(G=D_n\) (the dihedral group of order \(2n\)). For example, if \(G=S_n\), then \(n=2\) if and only if \(\mathcal R(\mathcal C)\) is simple if and only if \(\mathcal R(\mathcal C)\) is local if and only if \(\mathcal R(\mathcal C)\) is a field. Similarly, if \(G=A_n\), then \(n=3\) if and only if \(\mathcal R(\mathcal C)\) is simple if and only if \(\mathcal R(\mathcal C)\) is local if and only if \(\mathcal R(\mathcal C)\) is a field. Also, if \(G=D_n\), and \(\mathcal J\) denotes the Jacobson radical of \(\mathcal R(\mathcal C)\), then \(\mathcal R(\mathcal C)/J\cong[\oplus_{i=1}^t\mathbb Z_{p_i}]\oplus\mathbb Z_2^n\), where \(p_1,p_2,\ldots,p_t\) are the prime divisors of \(n\).NEWLINENEWLINE The paper concludes with the case where \(G\) is a finite nilpotent group. Because of the direct sum decomposition \(G=S(p_1)\oplus\cdots\oplus S(p_N)\) into Sylow subgroups, it suffices to focus on \(p\)-groups only. The following is shown: If \(G\) is a finite \(p\)-group, then \(\mathcal R(\mathcal C)\) is semisimple if and only if \(G\) has exponent \(p\) if and only if \(\mathcal R(\mathcal C)\cong\mathbb Z_p^n\), where \(n\) is the number of subgroups of \(G\) of order \(p\). Still for the \(p\)-group case: \(\mathcal R(\mathcal C)\) is local if and only if \(G\) has a unique subgroup of order \(p\) if and only if \(G\) is cyclic or generalized quaternion.