THE SYMMETRIC GENUS OF 2-GROUPS

Let G be a finite group. The symmetric genus sigma(G) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2(m) that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2(m-3) and a dihedral subgroup of index 4 or else the exponent of G is 2(m-2) . We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G. A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).