# Difference between revisions of "Brauer's permutation lemma"

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Brauer's permutation lemma has the following equivalent forms: | 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 | * 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 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 | * 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 |

## Revision as of 22:59, 16 September 2007

## 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