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→Group structure: normalizer |
→Subgroups: Weyl group |
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* The subgroup where all entries are ±1 is the [[signed permutation matrices]], which is the [[hyperoctahedral group]].
* The subgroup where the entries are ''m''th [[roots of unity]] <math>\mu_m</math> is isomorphic to a [[generalized symmetric group]].
* The subgroup of diagonal matrices is abelian, normal, and a maximal abelian subgroup. The quotient group is the symmetric group, and this construction is in fact the [[Weyl group]] of the general linear group: the diagonal matrices are a [[maximal torus]] in the general linear group (and are their own centralizer), the generalized permutation matrices are the normalizer of this torus, and the quotient, <math>N(T)/Z(T) = N(T)/T \cong S_n</math> is the Weyl group.
== Properties ==
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