Order statistics of a finite group determine whether it is nilpotent

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Statement

Whether or not a finite group is nilpotent is determined completely by its order statistics.

More specifically, consider a finite group of order n. This group is nilpotent if and only if, for every prime p dividing the order of n, the number of elements of the group whose order is a power of p is equal to the largest power of p dividing n.

In particular, a finite nilpotent group cannot be order statistics-equivalent a finite group that is not nilpotent.