The real genus of the alternating groups.
A Klein surface with boundary of algebraic genus \(\mathfrak p\geq 2\), has at most \(12(\mathfrak p-1)\) automorphisms. The groups attaining this upper bound are called \(M^*\)-groups, and the corresponding surfaces are said to have maximal symmetry. The \(M^*\)-groups are characterized by a partial presentation by generators and relators. The alternating groups \(A_n\) were proved to be \(M^*\)-groups when \(n\geq 168\) by M. Conder. In this work the authors prove that \(A_n\) is an \(M^*\)-group if and only if \(n\geq 13\) or \(n=5\), \(10\). In addition, they describe topologically the surfaces with maximal symmetry having \(A_n\) as automorphism group, in terms of the partial presentation of the group. As an application they determine explicitly all such surfaces for \(n\leq 14\). Each finite group \(G\) acts as an automorphism group of several Klein surfaces. The minimal genus of these surfaces is called the real genus of the group, \(\rho(G)\). If \(G\) is an \(M^*\)-group then \(\rho(G)=\tfrac{o(G)}{12}+1\).
scientific article; zbMATH DE number 5509265
scientific article