Brauer's permutation lemma

Statement

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