  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
$X=(x_{hg^{-1}})_{h\in H,g\in G}$, where $x_g$ ($g\in G$) are indeterminates
indexed by the elements in $G$. Then, we define an invariant $\Theta(G,H)$ 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 $\Theta(G,H)$ 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.
