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→Signed permutation group: D_n is not in general the kernel of the determinant - it is defined by having an even number of negative elements. |
→Group structure: Notation of semidirect product corrected: The symmetric group acts on the group of diagonal matrices, not the other way around. |
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The abstract group of generalized permutation matrices is the [[wreath product]] of ''F''<sup>×</sup> and ''S''<sub>''n''</sub>. Concretely, this means that it is the [[semidirect product]] of Δ(''n'', ''F'') by the [[symmetric group]] ''S''<sub>''n''</sub>:
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where ''S''<sub>''n''</sub> acts by permuting coordinates and the diagonal matrices Δ(''n'', ''F'') are isomorphic to the ''n''-fold product (''F''<sup>×</sup>)<sup>''n''</sup>.
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