Maximal subnear-rings of functions. (Q999098)

For a group \(G\), \(M(G)\) and \(M_0(G)\) denote the near-rings of all self-maps and all zero-preserving self-maps on \(G\), respectively. When \(G\) is finite, the maximal subnear-rings of \(M_0(G)\) are known [cf. \textit{C. Neumaier}, Commun. Algebra 33, No. 8, 2499-2518 (2005; Zbl 1081.16050)]. Let \(E(G) \) denote the subnear-ring of \(M_0(G)\) generated by the endomorphisms of \(G\). When \(G\) is Abelian or finite, it is known exactly which groups \(G\) will have \(E(G)\) as a maximal subnear-ring in \(M_0(G)\) [see \textit{A. Kreuzer} and \textit{C. J. Maxson}, Forum Math. 18, No. 1, 107-114 (2006; Zbl 1100.20037) and \textit{C. J. Maxson} and \textit{M. R. Pettet}, Arch. Math. 88, No. 5, 392-402 (2007; Zbl 1125.16036)]. In this paper the natural extension of these problems to infinite groups is considered by the author. For an arbitrary group \(G\), a complete characterization of the maximal subnear-rings of both \(M(G)\) and \(M_0(G)\), respectively, is given. When \(G\) is an infinite nonabelian group, then a number of criteria on \(G\) is given to ensure that \(E(G)\) is not a maximal subnear-ring of \(M_0(G)\). A complete characterization of the groups \(G\) for which \(E(G)\) is a maximal subnear-ring of \(M_0(G)\) remains open.