# Difference between revisions of "Order statistics of a finite group"

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The '''order statistics''' of a [[finite group]] is a function <math>\mathbb{N}_0 \to \mathbb{N}_0</math> which takes <math>n</math> and outputs the number of elements <math>x</math> whose order is <math>n</math>. | The '''order statistics''' of a [[finite group]] is a function <math>\mathbb{N}_0 \to \mathbb{N}_0</math> which takes <math>n</math> and outputs the number of elements <math>x</math> whose order is <math>n</math>. | ||

− | If <math>f</math> denotes the order statistics function, then the Dirichlet convolution <math>F = f * U</math> gives, for each <math>n</math>, the number of elements <math>x</math> satisfying <math>x^n = e</math>. | + | If <math>f</math> denotes the order statistics function, then the [[number:Dirichlet convolution|Dirichlet convolution]] <math>F = f * U</math> gives, for each <math>n</math>, the number of elements <math>x</math> satisfying <math>x^n = e</math>. |

+ | |||

+ | Two finite groups that have the same order statistics are termed [[order statistics-equivalent finite groups]]. | ||

==Facts== | ==Facts== |

## Revision as of 22:26, 24 June 2009

*This article talks about a statistics, which could be a function or a set of numbers, associated with any finite group*

## Contents

## Definition

The **order statistics** of a finite group is a function which takes and outputs the number of elements whose order is .

If denotes the order statistics function, then the Dirichlet convolution gives, for each , the number of elements satisfying .

Two finite groups that have the same order statistics are termed order statistics-equivalent finite groups.

## Facts

The order statistics function for a group cannot be chosen arbitrarily. It is subject to some constraints.

### Number of nth roots is a multiple of n

`Further information: Number of nth roots is a multiple of n`

For any , the number of roots of the identity is a multiple of the gcd of and the order of the group.

### Number of elements of prime order is nonzero

`Further information: Cauchy's theorem`

For any prime dividing the order of the group, there is a cyclic subgroup of order . Hence, .