Generalized symmetric group
- For the generalized symmetric group is exactly the ordinary symmetric group:
- For one can consider the cyclic group of order 2 as positives and negatives () and identify the generalized symmetric group with the signed symmetric group.
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 (for m odd this is isomorphic to ): the 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 (concretely, by taking the product of all the values), while the sign map on the symmetric group yields the These are independent, and generate the group, hence are the abelianization.
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.
- Davies, J. W.; Morris, A. O. (1974), "The Schur Multiplier of the Generalized Symmetric Group" (PDF), J. London Math. Soc., 2, 8: 615–620
- Can, Himmet (1996), "Representations of the Generalized Symmetric Groups", Contributions to Algebra and Geometry, 37 (2): 289–307
- Osima, M. (1954), "On the representations of the generalized symmetric group", Math. J. Okayama Univ., 4: 39–54