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## Brauer's permutation lemma

, 23:15, 16 September 2007
Applications
Note that since the field has characteristic zero, the irreducible representations are the same as indecomposable representations.
Consider the matrix with rows indexed by indecomposable representations, columns indexed by conjugacy classes, and the entry in row $\rho$ and column $c$ is $\chi(c,\rho)$. In other words, the matrix is the [[character table]]. The automorphism group acts on the rows and on the columns in a way that the entries are not affected. {{further|[[Conjugacyclass-class representation duality]]}}
Applying Brauer's permutation lemma to this, we get that for any automorphism, the sizes of orbits of conjugacy classes under the automorphism, are the same as the sizes of orbits of irreducible representations. In particular, the number of invariant conjugacy classes equals the number of invariant irreducible representations.
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