The real genus of groups of odd order

Let G be a finite group. The real genus rho(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we consider groups of odd order acting on bordered surfaces. First we show that if G is a group of odd order, then the real genus rho(G) is even. We also obtain a stronger result for p-groups. Let p be an odd prime, and let G be a p-group with rho(G) : 2; then the real genus rho(G) equivalent to p + 1 mod 2p. We also examine "large" automorphism groups of odd order. If the odd order group G acts on a bordered Klein surface of genus g >= 2, then vertical bar G vertical bar <= 3(g - 1). If G acts with the largest possible order 3(g - 1), then we call G an O*-group. In general, a quotient Q of an O*-group G is again an O*-group, and a surface X on which G acts is a full covering of a surface of lower genus on which Q acts. Thus, it is natural to consider the notion of an O*-simple group, that is, an O*-group with no O*-quotient. We classify the O*-simple groups.