Covering spaces and enumeration of regular covering spaces

Abstract In this thesis the theory of regular coverings of surfaces is algebraically investigated, in particular we determine the number of non-equivalent regular coverings of a compact, connetted orientable (non-orientable and punctured) surface of genus g with a given finite coverings groups G. The main theorem concerning classification of regular covering surfaces stated that [ 3,15,16,19,] " the normal subgroup N of the fundamental group Ilg>r (or lig,.) of a compact, connected orientable (or non-orientable) surface J] (or ^] ) of genus g with r > 0 punctures are in one-to-one correspondence with the equivalent classes of regular coverings of ^ (or 2 ) "¦ By means of this theorem, the topological problem of determining the number of regular coverings of a compact, connected surface with a given finite covering group G is reduced completely to the algebraic problem of determining the number of normal subgroup of the fundamental group of this surface under this correspondence. The aim is to apply techniques from finite group theory such as character theory, representation theory and Philip Hall's algebraic version of the Möbius Inversion Formula to solve this algebraic problem and to study regular coverings of surfaces.