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

Representation theory

There is a natural representation of elements of S(m,n) 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 (concretely, the abelianization) is (for m odd this is isomorphic to Z_{2m}): the Z_m factors (which are all conjugate, hence must map identically in an abelian group, since conjugation is trivial in an abelian group) can be mapped to Z_m (concretely, by taking the product of all the Z_m values), while the sign map on the symmetric group yields the Z_2. These are independent, and generate the group, hence are the abelianization. The second homology group (in classical terms, the Schur multiplier) 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.