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Abstract
Abstract Let G be a finite group. The real genus ρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we obtain a good general lower bound for the real genus of the group G. We use the standard representation of G as a quotient of a non-euclidean crystallographic group by a bordered surface group. The lower bound is used to determine the real genus of several infinite families of groups; the lower bound is attained for some of these families. Among the groups considered are the dicyclic groups and some abelian groups. We also obtain a formula for the real genus of the direct product of an elementary abelian 2-group and an “even” dicyclic group. In addition, we calculate the real genus of an abstract family of groups that includes some interesting 3-groups. Finally, we determine the real genus of the direct product of an elementary abelian 2-group and a dihedral group.
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