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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

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