Generalized symmetric group


In mathematics, the generalized symmetric group is the wreath product of the cyclic group of order m and the symmetric group of order n.

Examples

There is a natural representation of elements of as generalized permutation matrices, where the nonzero entries are m-th roots of unity:
The representation theory has been studied since ; see references in. As with the symmetric group, the representations can be constructed in terms of Specht modules; see.

Homology

The first group homology group is : the factors can be mapped to , while the sign map on the symmetric group yields the These are independent, and generate the group, hence are the abelianization.
The second homology group is given by :
Note that it depends on n and the parity of m: and which are the Schur multipliers of the symmetric group and signed symmetric group.