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

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Suppose $G$ is a finite group and $r$ is an integer relatively prime to the order of $G$. Suppose $K$ is a splitting field of $G$ of the form $\mathbb{Q}(\zeta)$ where $\zeta$ is a primitive $d^{th}$ root of unity, with $d$ dividing the order of $G$ (this can always be found, because sufficiently large implies splitting). 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 $K$, denoted $I(G)$, induced by the Galois automorphism of $K$ that sends $\zeta$ to $\zeta^r$.