Numerical Campedelli Surfaces Cannot have the Symmetric Group as the Algebraic Fundamental Group

Abstract

The non-existence set forth in the title is proved. It is known that for numerical Campedelli surfaces the algebraic fundamental group is of order ⩽9 and that the dihedral group of order 8 cannot occur. Therefore the quaternionic group is the only non-abelian algebraic fundamental group in this range.