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

From Groupprops

(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...") |
|||

Line 2: | Line 2: | ||

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>. | ||

+ | |||

+ | ==Related facts== | ||

+ | |||

+ | ===Converse=== | ||

+ | |||

+ | * [[Degrees of irreducible representations need not determine conjugacy class size statistics]] | ||

+ | |||

+ | ===Similar facts=== | ||

+ | |||

+ | * [[Conjugacy class size statistics need not determine group up to isoclinism]] | ||

+ | * [[Conjugacy class size statistics need not determine nilpotency class]] | ||

+ | * [[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 and such that the conjugacy class size statistics of are the same as those of (i.e., the two groups have the same number of conjugacy classes of each size) but the Degrees of irreducible representations (?) over for are not the same as those of .

## Related facts

### Converse

### Similar facts

- Conjugacy class size statistics need not determine group up to isoclinism
- Conjugacy class size statistics need not determine nilpotency class
- Conjugacy class size statistics need not determine derived length

## 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.