The Real Genus of <i>p</i>-Groups

Abstract

Let G be a finite group. The real genus p(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. We show that a non-cyclic p-group G has real genus not congruent to (1 — p3) (mod 2p3) if and only if G is in one of 14 families of groups. The genus formula for each of these 14 families of groups is determined. A consequence of this classification is that almost all positive integers that are the real genus of a p-group are congruent to (1 p3) (mod 2p3). Finally, the integers that occur as the real genus of a p-group with Frattini-class 2 have density zero in the positive integers.