Kourovka 1.65

### What is the conjecture
**"Is the class of groups of abelian extensions of abelian groups closed under taking direct sums?"**

Here, the "group of abelian extensions of abelian groups" refers to the group **Ext(A, B)** (also denoted as Ext^1(A, B)), which consists of equivalence classes of abelian extensions of an abelian group $B$ by an abelian group $A$ under the Baer sum. The problem asks whether the class of all such groups (as we vary $A$ and $B$) is closed under the direct sum operation. In other words:
If $G$ and $H$ are groups such that $G \cong$ Ext(A, B) and $H \cong$ Ext(C, D) for some abelian groups $A, B, C, D$, is it always true that $G \oplus H \cong$ Ext(X, Y) for some abelian groups $X, Y$?

### Status of the Problem
The problem is **unsolved**.
As of the latest editions of the Kourovka Notebook (including the 20th edition and updates leading into the 21st edition in 2025/2026), Problem 1.65 is still listed in the "1st Issue (1965)" section without a solution or a comment indicating it has been solved. This suggests that the answer is not yet known, or at least no solution has been widely recognized and recorded in the Notebook.

* The structure of Ext(A, B) has been studied extensively by authors like **L. Fuchs**, **S. Mac Lane**, and others. It is known that Ext(A, B) is always a **cotorsion group** (a group $G$ such that Ext $\mathbb{Q}, G) = 0$).
* The class of cotorsion groups is closed under direct sums. However, not all cotorsion groups are of the form Ext(A, B).
* The difficulty of the problem lies in the fact that while Ext converts direct sums in the first variable to direct products and in the second variable to direct products (or sums, depending on the index set and argument), the direct sum of two arbitrary Ext groups might not structurally align with a single Ext group of some other pair, especially when dealing with infinite ranks and specific torsion properties.

### Prerequisites needed
To state Kourovka Problem 1.65 in Lean 4's Mathlib **exactly as phrased** ("groups of abelian extensions"), the primary missing or non-standard definition is the **explicit construction of the group of extensions via the Baer sum**.

While Mathlib has a robust library for **homological algebra** and defines `Ext` as a **derived functor** (specifically in `Algebra.Homology` and `CategoryTheory`), it likely lacks the concrete, "element-wise" definition of Ext^1(A, B) as the set of equivalence classes of short exact sequences $0 \to B \to E \to A \to 0$ equipped with the **Baer sum** as the group operation.

To state the question formally, you would need:

1. **The Group of Extensions (Baer Sum):** A definition `ExtensionGroup (A B : AddCommGroup)` that constructs the abelian group of extensions. In Mathlib, the preferred approach is abstract (derived functors), so the concrete "Baer sum" construction is likely missing or would need to be defined to match the problem's specific terminology strictly.
2. **Comparison Isomorphism (Optional but related):** If you use the existing derived functor `Ext` to state the problem, you are stating an *equivalent* mathematical problem, but not the *literal* one. To bridge the gap, you would need the theorem establishing the isomorphism between the derived functor `Ext^1` and the group of extensions (Yoneda extensions/Baer sum).
3. **Universe Polymorphism for "The Class of Groups":** The problem asks about the "class of groups." In Lean (Type Theory), you cannot simply have a "set of all groups." You would need to state the problem using universe levels, defining a predicate `is_ext_group` for groups in a specific universe, which is a standard structural requirement in Lean but not a "missing definition" per se.

**Summary of Missing Structures:**
* **`BaerSum`** (The concrete group operation on extensions).
* **`ExtensionGroup A B`** (The concrete type of extensions modulo equivalence).

If you are willing to substitute "Group of Abelian Extensions" with "The Derived Functor `Ext`", then **no major definitions are missing**, as Mathlib contains `AddCommGrp`, `DirectSum`, and the `Ext` functor (in `CategoryTheory.Abelian.Ext` or `Algebra.Homology`). You would simply state:
```lean
def IsExtGroup (G : AddCommGroup.{u}) : Prop :=
∃ (A B : AddCommGroup.{u}), Nonempty (G ≃+ ((Ext A B).app ...)) -- pseudo-code accessing the Ext group
```
and ask if this predicate is closed under `Prod` (direct sum).

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-16

### Choose either option
- [ ] I plan on adding this conjecture to the repository
- [X] This issue is up for grabs: I would like to see this conjecture added by somebody else