Generalized symmetric group

Definition

The generalized symmetric group with parameters $m$ and $n$, denoted $S(m,n)$, is defined in the following equivalent ways:

1. It is the external wreath product of the cyclic group $\mathbb {Z} _{m}$ of order $m$, and the symmetric group $S_{n}$ (specifically, symmetric group on finite set) of degree $n$, with the natural action of the latter on a set of size $n$.
2. It is the external semidirect product of the homocyclic group $(\mathbb {Z} _{m})^{n}$ and the symmetric group of degree $n$, where the latter has a natural action by coordinate permutations.
3. It is the subgroup of the general linear group $GL(n,\mathbb {C} )$ over the field of complex numbers comprising monomial matrices (i.e., matrices where every row has exactly one nonzero entry and every column has exactly one nonzero entry) where all the nonzero entries are $m^{th}$ roots of unity. Note that the group also has a faithful monomial representation of degree $n$ over any field where the polynomial $x^{m}-1$ splits completely.

Particular cases

Two very special cases

• In the case $n=1$, we get the usual symmetric group (specifically, the symmetric group on finite set).
• In the case $n=2$, we get the signed symmetric group.