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

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Statement

Suppose G is a finite group and r is an integer relatively prime to the order of G. Suppose K is a field and L is a splitting field of G of the form K(\zeta) where \zeta is a primitive d^{th} root of unity, with d also relatively prime to r (in fact, we can arrange d to divide the order of G because sufficiently large implies splitting). Suppose there is a Galois automorphism of L/K that sends \zeta to \zeta^r. Consider the following two permutations:

  • The permutation on the set of conjugacy classes of G, denoted C(G), induced by the mapping g \mapsto g^r.
  • The permutation on the set of irreducible representations of G over L, denoted I(G), induced by the Galois automorphism of L that sends \zeta to \zeta^r.

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.