Brauer's permutation lemma

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Statement

Brauer's permutation lemma has the following equivalent forms:

  • 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