Application of Brauer's permutation lemma to group automorphism on conjugacy classes and irreducible representations

Statement

Suppose $G$ is a finite group. Suppose $\sigma$ is an automorphism of $G$. Then, $\sigma$ induces permutations both on the set of conjugacy classes $C(G)$ of $G$ and on the set of equivalence classes of irreducible representations $I(G)$ of $G$ over a splitting field. Both these permutations have the same cycle type.

Related facts

Similar facts

• Application of Brauer's permutation lemma to Galois automorphism on conjugacy classes and irreducible representations
• 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 equals number of orbits under automorphism group
• Number of orbits of irreducible representations equals number of orbits of conjugacy classes under any subgroup of 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

Facts used

1. Brauer's permutation lemma