The Symmetric Genus of p-Groups

Abstract

Let G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts. We show that a non-cyclic p-group G has symmetric genus not congruent to 1(mod p ³) if and only if G is in one of 10 families of groups. The genus formula for each of these 10 families of groups is determined. A consequence of this classification is that almost all positive integers that are the genus of a p-group are congruent to 1(mod p ³). Finally, the integers that occur as the symmetric genus of a p-group with Frattini-class 2 have density zero in the positive integers.

Key Words:

• Density
• Fuchsian group
• Frattini-class 2
• p-Group
• Symmetric genus
• Riemann surface

2010 Mathematics Subject Classification:

• Primary: 20H10
• Secondary: 20D15
• 20F38
• 30F99
• 57M60

Notes

Communicated by A. Tarull