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

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

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, $f(p) \ge p-1$.

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.