On Klein Surfaces with 2p or 2p + 2 Automorphisms

It is known that the maximal order of a cyclic group of automorphisms admitted by a bordered Klein surface or real algebraic curve of algebraic genus p is 2p or 2(p + 1), depending on whether p is odd or even. In this paper, we classify the automorphism groups of all bordered Klein surfaces or real algebraic curves which admit an automorphism group of order 2p, if p is odd, or 2(p + 1), if p is even. The topological type of each of these surfaces is determined. The defining equations of these real algebraic curves are presented in all but two cases. For these cases examples are given.