Generalized permutation matrix: Difference between revisions

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Revision as of 04:31, 21 January 2006

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==Group theory==

The 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 nonsingular diagonal matrices Δ(''n'', ''F'') forms a [[normal subgroup]]. One can show that the group of ''n''×''n'' generalized permutation matrices is a [[semidirect product]] of Δ(''n'', ''F'') by the [[symmetric group]] ''S''_''n'':

:Δ(''n'', ''F'') ⋊ ''S''_''n''.

Since Δ(''n'', ''F'') is isomorphic to (''F''^×)^''n'' and ''S''_''n'' acts by permuting coordinates, this group is actually the [[wreath product]] of ''F''^× and ''S''_''n''.