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Craftylamb (talk | contribs) m →Group structure: I think this is not true over the finite field of order 2, but its true in all other fields |
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===Group structure===
The [[set (mathematics)|set]] of ''n'' × ''n'' generalized permutation matrices with entries in a [[field (mathematics)|field]] ''F'' forms a [[subgroup]] of the [[general linear group]] GL(''n'', ''F''), in which the group of [[invertible matrix|nonsingular]] diagonal matrices Δ(''n'', ''F'') forms a [[normal subgroup]]. Indeed, over all fields except [[GF(2)]], the generalized permutation matrices are the [[normalizer]] of the diagonal matrices, meaning that the generalized permutation matrices are the ''largest'' subgroup of GL(''n'', ''F'') in which diagonal matrices are normal.
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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