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

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(Created page with "==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 [[splitt...")
 
 
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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>\mathbb{Q}(\zeta)</math> where <matH>\zeta</math> is a primitive <math>d^{th}</math> root of unity, with <math>d</math> dividing the order of <math>G</math> (this can always be found, because [[sufficiently large implies splitting]]). Consider the following two permutations:
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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>K</math>, denoted <math>I(G)</math>, induced by the Galois automorphism of <math>K</math> that sends <math>\zeta</math> to <math>\zeta^r</math>.
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* 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 G is a finite group and r is an integer relatively prime to the order of G. Suppose K is a field and L is a splitting field of G of the form K(\zeta) where \zeta is a primitive d^{th} root of unity, with d also relatively prime to r (in fact, we can arrange d to divide the order of G because sufficiently large implies splitting). Suppose there is a Galois automorphism of L/K that sends \zeta to \zeta^r. 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 L, denoted I(G), induced by the Galois automorphism of L that sends \zeta to \zeta^r.

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.