# Order statistics of a finite group

From Groupprops

*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 .

## 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

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

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