Editing Generalized permutation matrix (section)

===Group structure===
The [[set (mathematics)|set]] of ''n''&thinsp;×&thinsp;''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>:
:''S''<sub>''n''</sub> ⋉ &Delta;(''n'', ''F''),
where ''S''<sub>''n''</sub> acts by permuting coordinates and the diagonal matrices Δ(''n'', ''F'') are [[group isomorphism|isomorphic]] to the ''n''-fold product (''F''<sup>×</sup>)<sup>''n''</sup>.

To be precise, the generalized permutation matrices are a (faithful) [[linear representation]] of this abstract wreath product: a realization of the abstract group as a subgroup of matrices.