Finite abelian groups with the same order statistics are isomorphic

Statement

Suppose  and  are Finite abelian group (?)s that are order statistics-equivalent: the order statistics of  equal the order statistics of . Then,  is isomorphic to .

Proof

Proof idea

We show that the invariants needed to describe a finite abelian group by the structure theorem are completely determined by the order statistics.

Proof details

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.