TITLE: Classification of subgroups of finitely generated abelian groups
QUESTION [10 upvotes]: A finitely generated abelian group $A$ is isomorphic to a direct sum of cyclic groups. I am interested in an extension of this result on couples of abelian groups $(A,B),$ where $B$ is a subgroup of $A.$ Consider the category of such couples $(A,B),$ where morphism $f:(A,B)\to (A',B')$ is a homomorphism $f:A\to A'$ such that $f(B)\subseteq B'.$ A couple $(A,B)$ is called cyclic if $A$ and $B$ are cyclic groups.
Question 1: Is it true that any couple of finitely generated abelian groups $(A,B)$ is isomorphic to a direct sum of cyclic couples?
If $A$ is free, it follows from the Smith normal form theorem. If there was a version of the Smith normal form theorem for arbitrary homomorphisms of finitely generated abelian groups, then, I believe, this result would follow.
Question 2: Is there a version of the Smith normal form theorem for arbitrary homomorphisms of finitely generated abelian groups?

REPLY [3 votes]: For finite abelian groups see Sapir, M. V.
Varieties with a finite number of subquasivarieties.
Sibirsk. Mat. Zh. 22 (1981), no. 6, 168–187, 226.