Order statistics of a finite group

Individual version
The order statistics of a finite group is a function $$\mathbb{N}_0 \to \mathbb{N}_0$$ which takes $$d$$ and outputs the number of elements $$x$$ whose order is $$d$$. 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 $$f$$ denotes the order statistics function, then the Dirichlet convolution $$F = f * U$$ gives, for each divisor $$d$$ of the order of the group, the number of elements $$x$$ satisfying $$x^d = e$$. 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
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, $$f(p) \ge p-1$$.

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

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


 * 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.