Application of Brauer's permutation lemma to Galois automorphism on conjugacy classes and irreducible representations
Suppose is a finite group and is an integer relatively prime to the order of . Suppose is a field and is a splitting field of of the form where is a primitive root of unity, with also relatively prime to (in fact, we can arrange to divide the order of because sufficiently large implies splitting). Suppose there is a Galois automorphism of that sends to . Consider the following two permutations:
- The permutation on the set of conjugacy classes of , denoted , induced by the mapping .
- The permutation on the set of irreducible representations of over , denoted , induced by the Galois automorphism of that sends to .
Then, these two permutations have the same cycle type. In particular, they have the same number of cycles, and the same number of fixed points, as each other.