Application of Brauer's permutation lemma to group automorphisms on conjugacy class-representation duality
Suppose is a finite group and is a splitting field for . Any automorphism of induces a permutation on the set of conjugacy classes of , and a permutation on the set of irreducible representations (up to equivalence) of .
Our statement is that the two permutations and have the same cycle type.
- Application of Brauer's permutation lemma to Galois automorphisms on conjugacy class-representation duality
- Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group
- Number of orbits of irreducible representations need not equal number of orbits of conjugacy classes under automorphism group
More on Brauer's permutation lemma
- Brauer's permutation lemma
- Analogue of Brauer's permutation lemma fails over rationals for every non-cyclic finite group
- Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals