Resumen de The symmetric genus of groups of odd order

Coy L. May, J. Zimmerman

• Let G be a finite group of odd order. The symmetric genus G is the minimum genus of any Riemann surface on which G acts faithfully. Suppose G acts on a Riemann surface X of genus g ³ 2. If |G| > 8(g - 1), then |G| = K(g-1), where K is 15, 21/2, 9 or 33/4. We call these four types of groups LO1-groups through LO4-groups, respectively. We determine the characteristics of LO1-groups and LO2-groups and show that there are infinite families of each type. We also show that there are exactly ten odd order groups with genus between 2 and 26 inclusive. Finally, if G is an odd order group with symmetric genus of the form p+1 for an odd prime p or 2k + 1, for some positive integer k, then G is a metacyclic group with certain properties. We determine that, in the range between 26 and 200, most of the numbers of either of these forms are not the genus of an odd order group.