A Reduction Theorem

Abstract

AbstractBy the Fundamental Theorem of Finite Abelian Groups, every finite Abelian group is a direct product of cyclic groups. We will show that the FT on a given finite Abelian group is the tensor product of the FT on the cyclic groups in its direct product decomposition. Thus, to understand the FT on finite Abelian groups, it suffices to investigate the FT on cyclic groups. This approach to reduction of the FT is to show that a similar reduction (or decomposition) holds for vector spaces of complex-valued functions on finite Abelian groups.