Finite abelian groups with the same order statistics are isomorphic
We show that the invariants needed to describe a finite abelian group by the structure theorem are completely determined by the order statistics.
By fact (1), any finite abelian group can be written as a direct product of cyclic groups of prime power order, with the number of copies of a cyclic group of prime power order independent of the choice of decomposition.
We claim that the order statistics of a finite group determine the number of times each cyclic group of prime power order occurs as a direct factor. First, for every prime , consider the subgroup of elements whose order is a power of (in other words, the -Sylow subgroup). This is obtained by grouping together, in the direct product decomposition, all cyclic groups of order a power of . Thus, we may restrict attention to this subgroup, so it suffices to consider the case that is a finite abelian -group.
For any , let be the number of cyclic group factors of order , let be the sum of the s, and let be the logarithm to base of the number of elements of order dividing . This makes sense since theelements of order dividing form a subgroup. Also, is clearly determined by the order statistics. We claim that the s determine the s.
An easy count shows that for all :
We thus get, taking a first difference:
Taking a second difference yields:
This shows that the s determine the s, completing the proof.