Difference between revisions of "Conjugacy class size statistics need not determine degrees of irreducible representations"

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(Created page with "==Statement== It is possible to have two finite groups <math>G_1</math> and <math>G_2</math> such that the [[fact about::conjugacy class size statistics of a finite group|co...")
 
 
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It is possible to have two [[finite group]]s <math>G_1</math> and <math>G_2</math> such that the [[fact about::conjugacy class size statistics of a finite group|conjugacy class size statistics]] of <math>G_1</math> are the same as those of <math>G_2</math> (i.e., the two groups have the same number of conjugacy classes of each size) but the [[fact about::degrees of irreducible representations]] over <math>\mathbb{C}</math> for <math>G_1</math> are not the same as those of <math>G_2</math>.
 
It is possible to have two [[finite group]]s <math>G_1</math> and <math>G_2</math> such that the [[fact about::conjugacy class size statistics of a finite group|conjugacy class size statistics]] of <math>G_1</math> are the same as those of <math>G_2</math> (i.e., the two groups have the same number of conjugacy classes of each size) but the [[fact about::degrees of irreducible representations]] over <math>\mathbb{C}</math> for <math>G_1</math> are not the same as those of <math>G_2</math>.
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==Related facts==
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===Converse===
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* [[Degrees of irreducible representations need not determine conjugacy class size statistics]]
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===Similar facts===
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* [[Conjugacy class size statistics need not determine group up to isoclinism]]
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* [[Conjugacy class size statistics need not determine nilpotency class]]
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* [[Conjugacy class size statistics need not determine derived length]]
  
 
==Proof==
 
==Proof==

Latest revision as of 04:08, 9 February 2013

Statement

It is possible to have two finite groups G_1 and G_2 such that the conjugacy class size statistics of G_1 are the same as those of G_2 (i.e., the two groups have the same number of conjugacy classes of each size) but the Degrees of irreducible representations (?) over \mathbb{C} for G_1 are not the same as those of G_2.

Related facts

Converse

Similar facts

Proof

Further information: Linear representation theory of groups of order 128#Degrees of irreducible representations, element structure of groups of order 128#Conjugacy class sizes

The smallest examples are among groups of order 128.