Order statistics of a finite group

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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.

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.

Other facts