Resumen de The Groups of Strong Symmetric Genus 4

J. Zimmerman, Coy L. May

• Let G be a finite group. The strong symmetric genus is the minimum genus of any Riemann surface on which G acts preserving orientation. The groups of strong symmetric genus 3 or less 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 2-groups 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.