The Groups of Strong Symmetric Genus 4 Coy L. May and Jay Zimmerman

Let G be a finite group. The strong symmetric genus σ0(G) is the minimum genus of any Riemann surface on which G acts preserving orientation. The groups of strong symmetric genus σ0 ≤ 3 have been classified. Here we classify the groups of strong symmetric genus four. There are exactly ten such groups; eight of these are automorphism groups of regular maps of genus 4. We also consider non-abelian p-groups that have an element of maximal possible order. We complete the determination of the strong symmetric genus of each p-group with this property. Conversely, the non-abelian 2groups of even positive strong symmetric genus have an element of maximum possible order. Further, we establish that for an odd prime p, the strong symmetric genus of a non-abelian p-group is congruent to one modulo a power of p.