Content deleted Content added
→Group theory: "left normal factor" instead of "right normal factor" sign for semidirect product (see semidirect product) |
m Corrected unicode character from html to \{\{unicode|⋉\}\} |
||
Line 16:
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''<sub>''n''</sub>:
:Δ(''n'', ''F'') {{unicode|⋉}} ''S''<sub>''n''</sub>.
Since Δ(''n'', ''F'') is isomorphic to (''F''<sup>×</sup>)<sup>''n''</sup> and ''S''<sub>''n''</sub> acts by permuting coordinates, this group is actually the [[wreath product]] of ''F''<sup>×</sup> and ''S''<sub>''n''</sub>.
| |||