Character groups of dihedral and generalized quaternion groups

Finite abelian group duality appears in the discrete Fourier transform, or as the finite fragment of Pontryagin duality. The dual or character group of an abelian group encodes the products of its (linear) irreducible characters. Now, recent developments in combinatorics enable the construction of character groups for finite dihedral and generalized quaternion groups, encoding the products of all the irreducible characters (linear and nonlinear) in purely multiplicative fashion. In particular, just as in the abelian case, each dihedral group may serve as its own character group. Furthermore, certain Adams operations on the characters are shown to correspond to powers in the character group of a dihedral group.