The aim of the present paper is to generalize the notion of the group
determinants for finite groups. For a finite group G of order kn and its
subgroup H of order n, one may define an n by kn matrix
X=(xhgβ1β)hβH,gβGβ, where xgβ (gβG) are indeterminates
indexed by the elements in G. Then, we define an invariant Ξ(G,H) for
a given pair (G,H) by the k-wreath determinant of the matrix X, where k
is the index of H in G. The k-wreath determinant of n by kn matrix is
a relative invariant of the left action by the general linear group of order
k and right action by the wreath product of two symmetric groups of order k
and n. Since the definition of Ξ(G,H) is ordering-sensitive,
representation theory of symmetric groups are naturally involved. In this
paper, we treat abelian groups with a special choice of indeterminates and give
various examples of non-abelian group-subgroup pairs.Comment: 12 pages, 2 figure