Abstract
LetGbe a finite abelian group, andYbe a closed surface. The problems of classifying and enumerating the free and effectiveG-actions onYmodulo selfhomeomorphisms ofYandX=Y/Gcan be transferred into ones of classifying regularG-coverings onX. P. A. Smith [7], proved that for any prime numberpthere arepr(r–1)/2equivalence classes of free (ℤp)ractions onYprovided thatrℤgenus ofX. This paper is devoted to the classification and the enumeration of regularG-covering surfaces, whenGis any finite abelian group. Recently, A. Edmonds [2] classified theG-actions on closed surfaces by theirG-bordism classes in the set(G) of free orientedG-cobordism classes of free orientedG-surfaces.
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