The real genus of cyclic by dihedral and dihedral by dihedral groups

Abstract

Every finite group G acts as an automorphism group of several bordered compact Klein surfaces. The minimal genus of these surfaces is called the real genus and denoted by ρ(G). The systematical study was begun by C.L. May and continued by him in several other papers about the topic. As a consequence of these works, he and other authors obtained the groups such that 0⩽ρ(G)⩽8. The real genus of many families of groups has also been calculated. In this work we are interested in finding the real genus of each groupDr×Ds, where both factors are dihedral groups. Results depend on the real genus of groupsCm×Dn, where Cm is a cyclic group. The case m odd was studied by May and the authors have studied the case m even. The result of May needs to be slightly corrected. In this work we complete the proof of May for the case m odd and we calculate the real genus of the groupsDr×Ds.