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

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## Statement

Suppose is a finite group and is an integer relatively prime to the order of . Suppose is a splitting field of of the form where is a primitive root of unity, with dividing the order of (this can always be found, because sufficiently large implies splitting). 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.