  Finite (or Discrete) Fourier Transforms (FFT) are essential tools in
engineering disciplines based on signal transmission, which is the case in most
of them. FFT are related with circulant matrices, which can be viewed as group
matrices of cyclic groups. In this regard, we introduce a generalization of the
previous investigations to the case of finite groups, abelian or not. We make
clear the points which were not recognized as underlying algebraic structures.
Especially, all that appears in the FFT in engineering has been elucidated from
the point of view of linear representations of finite groups. We include many
worked-out examples for the readers in engineering disciplines.

Number Theory;Commutative Algebra