In mathematics, the generalized symmetric group is the wreath product S ( m , n ) := Z m ≀ S n of the cyclic group of order m and the symmetric group of order n.
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 ≅ { ± 1 } ) and identify the generalized symmetric group S ( 2 , n ) with the signed symmetric group.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 ≅ μ m .
The representation theory has been studied since (Osima 1954); see references in (Can 1996). As with the symmetric group, the representations can be constructed in terms of Specht modules; see (Can 1996).
The first group homology group (concretely, the abelianization) is Z m × Z 2 (for m odd this is isomorphic to Z 2 m ): 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 (Davies & Morris 1974):
H 2 ( S ( 2 k + 1 , n ) ) = { 1 n < 4 Z / 2 n ≥ 4. H 2 ( S ( 2 k + 2 , n ) ) = { 1 n = 0 , 1 Z / 2 n = 2 ( Z / 2 ) 2 n = 3 ( Z / 2 ) 3 n ≥ 4. Note that it depends on n and the parity of m: H 2 ( S ( 2 k + 1 , n ) ) ≈ H 2 ( S ( 1 , n ) ) and H 2 ( S ( 2 k + 2 , n ) ) ≈ H 2 ( S ( 2 , n ) ) , which are the Schur multipliers of the symmetric group and signed symmetric group.
