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
* [[Analogue of Brauer's permutation lemma fails over rationals for every non-cyclic finite group]]: If <math>G</math> is a non-cyclic finite group, we can find two permutation representations <math>\varphi_1, \varphi_2</math> of <math>G</math> that are equivalent as linear representations over the rational numbers but not as permutation representations.