# Classifying maps of finitely generated abelian groups up to automorphism

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?

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.

• Wow, I seriously underestimated this. I thought if nobody answers I'll work it out for myself! Thanks for saving me from that! Commented Jul 2 at 10:52