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 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.
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.
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