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

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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 automorphisms have the same cycle type.

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