Order statistics of a finite group determine whether it is nilpotent
Whether or not a finite group is nilpotent is determined completely by its order statistics.
More specifically, consider a finite group of order . This group is nilpotent if and only if, for every prime dividing the order of , the number of elements of the group whose order is a power of is equal to the largest power of dividing .
In particular, a finite nilpotent group cannot be order statistics-equivalent a finite group that is not nilpotent.