Brauer's permutation lemma
= Brauer's permutation lemma
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A group of <automorphisms> of a <finite group> permutes both its <conjugacy classes> and its <irreducible characters>. The two <permutation representations> have the same <character>: each <automorphism> fixes equally many <conjugacy classes> and irreducible characters. The invertible <character table> intertwines the two permutation <matrices>, which have equal traces. The claim concerns fixed classes, not merely fixed elements.