# Order statistics of a finite group

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

## Contents

## Definition

### Individual version

The **order statistics** of a finite group is a function which takes and outputs the number of elements whose order is . The function is usually restricted only to divisors of the order of the group, because order of element divides order of group.

### Cumulative version

If denotes the order statistics function, then the Dirichlet convolution gives, for each divisor of the order of the group, the number of elements satisfying . This function is termed the **cumulative order statistics** function. The order statistics can be deduced from the cumulative order statistics and vice versa.

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, .

## Relation with arithmetic functions

The order statistics of a finite group can be used to deduce the values of the following arithmetic function:

Arithmetic function | Meaning | How it can be deduced from the order statistics |
---|---|---|

order | the number of elements in the group | add up the number of elements of each order |

exponent | the lcm of the orders of all elements | take the lcm of all elements that give nonzero output under the order statistics function. |

## Relation with group properties

A better tabulated version of this information is available at order statistics-equivalent finite groups#Relation with group properties.

- Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring
- Order statistics of a finite group determine whether it is nilpotent
- Finite abelian groups with the same order statistics are isomorphic

## GAP implementation

GAP code can be written to output the order statistics of any finite group in a number of convenient formats. See the code for some such functions at GAP:OrderStatistics and GAP:CumulativeOrderStatistics.