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Brauer's permutation lemma

151 bytes added, 03:51, 14 January 2014
Facts
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 [[cycle type]] i.e., the same number of cycles of a given each lengthin their [[cycle decomposition]]s.
# 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 [[cycle type]] (i.e., the same number of cycles of each lengthin their [[cycle decomposition]]s), vizi.e., 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
==Facts==
 
===Opposite facts===
* [[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.
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