Classifying maps of finitely generated abelian groups up to automorphism

Question

We have a nice characterization of f.g. abelian groups that factors them into cyclic groups. A homomorphism between them is just a matrix of maps $\mathbb{Z}_{p^i} \to \mathbb{Z}_{p^j}$ between the factors. I would like to classify these homomorphisms $\varphi$ up to composing automorphisms $\alpha \circ \varphi \circ\beta$ on both sides, which is analogous to classifying matrices up to matrix equivalence. This should produce some restricted rank normal form / Smith normal form matrices. Are there existing literature on this classification?

Answer 1

This is a very natural question, but the answer is that such a classification is almost certainly hopeless.

In this answer of mine to a related MathOverflow question, I give a reference to the paper

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

in which it is proved that the classification up to isomorphism of pairs $(G,H)$, where $G$ is a finite abelian group of exponent $p^n$ (for $p$ a fixed prime) and $H\leq G$ is a subgroup, is in some sense a "wild problem" if $n\geq7$. The original meaning of "wild" in this context was for classifying modules for finite-dimensional algebras over algebraically closed fields, but Ringel and Schmidmeier explain in the paper exactly what they mean here. But roughly, it is believed that, for any field $k$, classifying pairs of square matrices over $k$ up to simultaneous conjugacy is a hopeless problem, and here we would take $k=\mathbb{F}_p$, the field with $p$ elements.

Now, if you could classify maps $\varphi:H\to G$ of finitely generated abelian groups up to isomorphism, then you could classify the injective ones, and in particular the injective ones where the exponent of $G$ is bounded by $p^7$. So this means that the classification of maps is at least as hard as the problem that Ringel and Schmidmeier consider.

To classify all maps, not just injective ones, I think that a lower exponent of $p^5$ for $G$ and $H$ is probably enough to give a wild problem. Some relevant references are

Brenner, Sheila, Large indecomposable modules over a ring of $2\times 2$ triangular matrices, Bull. Lond. Math. Soc. 3, 333-336 (1971). ZBL0223.16012,

and

Skowroński, Andrzej, Tame triangular matrix algebras over Nakayama algebras, J. Lond. Math. Soc., II. Ser. 34, 245-264 (1986). ZBL0606.16021.