Analogue of Brauer's permutation lemma fails over rationals for every non-cyclic finite group
Let be a finite group that is not a cyclic group (in particular, not a finite cyclic group). Then, there exists two permutation representation such that and are equivalent as linear representations over (viewed this way by the embedding of in as permutation matrices) but are not equivalent as permutation representations.
In other words, the analogue of Brauer's permutation lemma fails to hold for non-cyclic finite groups.
- Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals
- Brauer's permutation lemma
- Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group