In this paper, we deal with a finite abelian group G and the abstract Fourier transform  Then we consider  where is defined by the composition with a bijection  is a pullback of j.) In particular, we show that  is a permutation if and only if j is a group isomorphism. Then we study how the spectrum of  depends on the isomorphism j.

finite abelian group, character, Fourier transform, Gauss sum, orthogonality relations, Fourier inversion.