# 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

## Definition

The order statistics of a finite group is a function $\mathbb{N}_0 \to \mathbb{N}_0$ which takes $n$ and outputs the number of elements $x$ whose order is $n$.

If $f$ denotes the order statistics function, then the Dirichlet convolution $F = f * U$ gives, for each $n$, the number of elements $x$ satisfying $x^n = e$.

## 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 $n$, the number $F(n)$ of $n^{th}$ roots of the identity is a multiple of the gcd of $n$ and the order of the group.

### Number of elements of prime order is nonzero

Further information: Cauchy's theorem

For any prime $p$ dividing the order of the group, there is a cyclic subgroup of order $p$. Hence, $f(p) \ge p-1$.