Originally Posted by

**Plato** TPH, I really wish that I could help you. It is clear to me that at this point you know far more about this then I do now. You see, I have not thought about this for years and years: not since I finish prelims in algebra. But I did go back and reread the section in Kurosh, the group theory text we used. It appears you have it correct. I will quote his summation.

“If a finitely generated Abelian group is decomposed into the direct sum of indecomposable summands, then the number of infinite cyclic summands and the totality of the orders of the primary cyclic summands is independent of the decomposition, that is, of the choice of a basis.

In other words, any two decompositions of a finitely generated Abelian into the direct sum of indecomposable cyclic groups are isomorphic.”