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-abelia...

Fix an integer n = 3. We show that the alternating group An appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same when n is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in an Sn–extension (i.e. a Galois extension with the symmetric group Sn as Galois group)...