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Brauer's permutation lemma

89 bytes added, 00:10, 10 April 2010
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Brauer's permutation lemma has the following equivalent forms:
* # If a row permutation and a column permutation have the same effect on a nonsingular matrix, then they must have the same number of cycles of a given length* # The [[symmetric group]] is a [[conjugacy-closed subgroup]] in the [[general linear group]] over any [[field]] of characteristic zero* # If two permutation matrices are conjugate in the general linear group over a field of characteristic zero, then they have the same number of cycles of each length, viz, are conjugate in the symmetric group itself* # If two permutation representations of a [[cyclic group]] are conjugate in the general linear group over a field of characteristic zero, they are also conjugate in the symmetric group ==Facts== * [[Symmetric group is not subset-conjugacy-closed in general linear group]]: In particular, in formulation (4), we ''cannot'' replace cyclic group by an arbitrary [[finite group]].
Notice that in the last formulation we ''cannot'' replace cyclic group by an arbitrary [[finite group]].
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