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

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==Statement== | ==Statement== | ||

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

* The permutation on the set of conjugacy classes of <math>G</math>, denoted <math>C(G)</math>, induced by the mapping <math>g \mapsto g^r</math>. | * The permutation on the set of conjugacy classes of <math>G</math>, denoted <math>C(G)</math>, induced by the mapping <math>g \mapsto g^r</math>. | ||

− | * The permutation on the set of irreducible representations of <math>G</math> over <math> | + | * The permutation on the set of irreducible representations of <math>G</math> over <math>L</math>, denoted <math>I(G)</math>, induced by the Galois automorphism of <math>L</math> that sends <math>\zeta</math> to <math>\zeta^r</math>. |

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. | 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. |

## Latest revision as of 18:56, 9 May 2011

## Statement

Suppose is a finite group and is an integer relatively prime to the order of . Suppose is a field and is a splitting field of of the form where is a primitive root of unity, with also relatively prime to (in fact, we can arrange to divide the order of because sufficiently large implies splitting). Suppose there is a Galois automorphism of that sends to . Consider the following two permutations:

- The permutation on the set of conjugacy classes of , denoted , induced by the mapping .
- The permutation on the set of irreducible representations of over , denoted , induced by the Galois automorphism of that sends to .

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.