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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 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 xn = e.
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 nth 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, 
.
Other facts
- 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
