SOME RESULTS ON O*-GROUPS

A compact Klein surface with boundary of algebraic genus g >= 2 has at most 12(g - 1) automorphisms. When a surface attains this bound, it has maximal symmetry, and the group of automorphisms is then called an M*-group. If a finite group G of odd order acts on a bordered Klein surface X of algebraic genus g >= 2, then vertical bar G vertical bar <= 3(g - 1). If G acts with the largest possible order 3(g - 1), then G is called an O*-group. In this paper, using the results about some normal subgroups of the extended modular group (Gamma) over bar, we obtain some results about O*-groups. Also, we give the relationships between O*-groups and M*-groups.