# Generalized symmetric group

In mathematics, the generalized symmetric group is the wreath product $S(m,n):=Z_{m}\wr S_{n}$ of the cyclic group of order m and the symmetric group on n letters.

## Representation theory

The representation theory has been studied since Template:Harv; see references in Template:Harv. As with the symmetric group, the representations can be constructed in terms of Specht modules; see Template:Harv.

## Homology

The first group homology group (concretely, the abelianization) is $Z_{m}\times Z_{2}$ (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 Template:Harv:

$H_{2}(S(2k+1,n))={\begin{cases}1&n<4\\\mathbf {Z} /2&n\geq 4.\end{cases}}$ $H_{2}(S(2k+2,n))={\begin{cases}1&n=0,1\\\mathbf {Z} /2&n=2\\(\mathbf {Z} /2)^{2}&n=3\\(\mathbf {Z} /2)^{3}&n\geq 4.\end{cases}}$ Note that it depends on n and the sign of m: $H_{2}(S(2k+1,n))\approx H_{2}(S(1,n))$ and $H_{2}(S(2k+2,n))\approx H_{2}(S(2,n)),$ which are the Schur multipliers of the symmetric group and signed symmetric group.