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 of order n.

Examples

 * For $$m=1,$$ the generalized symmetric group is exactly the ordinary symmetric group: $$S(1,n) = S_n.$$
 * For $$m=2,$$ one can consider the cyclic group of order 2 as positives and negatives ($$Z_2 \cong \{\pm 1\}$$) and identify the generalized symmetric group $$S(2,n)$$ with the signed symmetric group.

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: $$Z_m \cong \mu_m.$$

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 $$Z_m \times Z_2$$ (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 :
 * $$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}$$ Note that it depends on n and the parity of m: $$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.