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

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

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