# Classification of subgroups of finitely generated abelian groups

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?

• A question in Math.SE is linking to this. I thought cross-linking would be ok in case there is more interest. Jul 23 '20 at 10:02

The answer to Question 1 is no.

Let $$A=\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$$ and let $$B$$ be the subgroup generated by $$(2,1)$$.

Since $$B$$ is cyclic of order $$4$$, if it were contained in a proper direct summand of $$A$$ then it would be contained in a cyclic subgroup of $$A$$ of order $$8$$, and so $$(2,1)$$ would be equal to $$2a$$ for some $$a\in A$$, which it's not.

In fact, if you consider only the case where $$A$$ has exponent dividing $$p^n$$ for some prime $$p$$ and natural number $$n$$, there are infinitely many indecomposable such couples if $$n=6$$, and the classification is in some sense "wild" if $$n\geq7$$. See, for example,

Ringel, Claus Michael; Schmidmeier, Markus, Submodule categories of wild representation type., J. Pure Appl. Algebra 205, No. 2, 412-422 (2006). ZBL1147.16019.

and its references.

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.