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
The order statistics of a finite group is a function which takes and outputs the number of elements whose order is .
If denotes the order statistics function, then the Dirichlet convolution gives, for each , the number of elements satisfying .
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 , the number of roots of the identity is a multiple of the gcd of and the order of the group.
Number of elements of prime order is nonzero
For any prime dividing the order of the group, there is a cyclic subgroup of order . Hence, .