Kourovka 1.67

### What is the conjecture
**"Determine the isomorphism class of the abelian group $N/[F, N]$ in terms of invariants of the group $G$."** Based on homological algebra, the answer appears to be the direct sum of the Schur multiplier $H_2(G)$ and a free abelian group of rank $d(G) - \text{rank}(G_{ab})$. (Gemini)

The problem posed in **Kourovka Notebook Problem 1.67** (attributed to **L. Ya. Kulikov**) is:

> "Suppose that $G$ is a finitely presented group, $F$ a free group whose rank is equal to the minimal number of generators of $G$, with a fixed homomorphism of $F$ onto $G$ with kernel $N$. Find a complete system of invariants of the factor-group of $N$ by the commutator subgroup $[F, N]$."

Proving that the group is isomorphic to a specific structure (like the one involving the Schur multiplier and d(G)) would constitute a major step, potentially "solving" it if the proof holds in the generality requested.

Despite being listed as unsolved, the group $N/[F, N]$ can be described using standard tools from **homological algebra**, specifically the theory of central extensions and the **Schur multiplier**.

1. **Abelian Structure**: Since $N$ is a normal subgroup of $F$, the commutator subgroup $[N, N]$ is contained in $[F, N]$. Consequently, the quotient group $N/[F, N]$ is **abelian**.
2. **Homological Sequence**: The group fits into the **five-term exact sequence** of homology associated with the extension $1 \to N \to F \to G \to 1$:
$$ H_2(F) \to H_2(G) \to N/[F, N] \to F_{ab} \to G_{ab} \to 0 $$
Since $F$ is a free group, $H_2(F) = 0$. The sequence becomes:
$$ 0 \to H_2(G) \to N/[F, N] \to \ker(F_{ab} \to G_{ab}) \to 0 $$
3. **Splitting**: The term $K = \ker(F_{ab} \to G_{ab})$ is a subgroup of the free abelian group $F_{ab} \cong \mathbb{Z}^{d(G)}$, so $K$ is a **free abelian group** of rank $d(G) - \text{rank}(G_{ab})$.
Since $K$ is free abelian, the short exact sequence splits. Therefore, as an abstract abelian group:
$$ N/[F, N] \cong H_2(G) \oplus \mathbb{Z}^{d(G) - \text{rank}(G_{ab})} $$
Here, $H_2(G)$ is the **Schur multiplier** of $G$.

### Why is it considered a problem?
Given the derivation above, the isomorphism class of $N/[F, N]$ is determined by the **Schur multiplier** $H_2(G)$, the **minimal number of generators** $d(G)$, and the rank of the abelianization $G_{ab}$.
The fact that it remains an open problem in the Kourovka Notebook suggests that:
* The "complete system of invariants" might be sought in a form that is more **explicitly computable** from the presentation, as $d(G)$ is generally uncomputable.
* There may be subtleties regarding the **dependence on the specific minimal presentation** (though the formula suggests invariance).
* The question might be interpreted as asking for invariants of the **embedding** of $N/[F,N]$ or its structure as a module (though the action is trivial), or perhaps the question is simply to prove that these *are* the complete invariants in a rigorous setting which hasn't been formally "checked off" in the Notebook's archives.

### Prerequisites needed
To state the conjecture (that $N/[F, N] \cong H_2(G) \oplus \mathbb{Z}^{d(G) - \text{rank}(G_{ab})}$) in Lean 4, you would need to define several concepts that are not currently available as single-line commands in Mathlib.

Here are the specific missing or required definitions:

1. **Minimal Number of Generators ($d(G)$)**
Mathlib has `Subgroup.closure` and finitely generated groups, but it does not have a computable or non-computable definition for the integer value $d(G)$.
* *Requirement:* A definition that returns the minimum cardinality of a generating set for a finitely generated group.
```lean
noncomputable def min_generators (G : Type*) [Group G] [FinitelyGeneratedGroup G] : ℕ := ...
```

2. **Torsion-Free Rank of an Abelian Group**
Mathlib has `Module.rank` (returning a `Cardinal`), but getting the specific integer rank for a finitely generated abelian group requires unwrapping the structure theorem or defining the dimension of the tensor product with $\mathbb{Q}$.
* *Requirement:* A definition returning the rank as a natural number.
```lean
noncomputable def torsion_free_rank (A : Type*) [AddCommGroup A] : ℕ := ...
```

3. **The Schur Multiplier ($H_2(G, \mathbb{Z})$)**
While Mathlib has general homological algebra and group cohomology (`RepresentationTheory.GroupCohomology`), instantiating $H_2(G, \mathbb{Z})$ specifically as a concrete `AddCommGroup` requires setting up the trivial representation and invoking the cohomology functors.
* *Requirement:* A clear definition of the second homology group.
```lean
def SchurMultiplier (G : Type*) [Group G] : Type* := GroupHomology G (Rep.trivial ℤ) 2
```

### Formal Statement of the Conjecture in Lean 4
With those definitions assumed (`min_gen`, `int_rank`, `SchurMult`), the conjecture can be stated as:

```lean
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.GroupTheory.Finiteness
import Mathlib.Algebra.Homology.GroupCohomology

-- Assuming the definitions discussed above exist:
variable (min_gen : Type* → ℕ)
variable (int_rank : Type* → ℕ)
variable (SchurMultiplier : Type* → Type*) [∀ G, AddCommGroup (SchurMultiplier G)]

theorem kourovka_1_67_conjecture
(G : Type*) [Group G] [FinitelyPresentedGroup G]
(F : Type*) [Group F] [FreeGroup F]
-- Condition: F has rank equal to d(G)
(h_rank : FreeGroup.rank F = min_gen G)
-- Condition: Fixed homomorphism onto G with kernel N
(φ : F →* G) (h_surj : Function.Surjective φ)
(N : Subgroup F) (h_ker : N = MonoidHom.ker φ) :
-- The Conjecture: The group N/[F, N] is isomorphic to H₂(G) ⊕ ℤ^(d(G) - rank(G_ab))
Nonempty (
(N ⧸ (commutator N (⊤ : Subgroup F))) ≃+
(SchurMultiplier G) ×
(Multiplicative (Fin (min_gen G - int_rank (Abelianization G)) → ℤ))
)
```

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

* ams-20

### 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

Start by reviewing the cited Mathlib imports and the proposed `kourovka_1_67_conjecture` statement. Check which group homology, finitely presented group, quotient, and rank definitions are actually available before deciding the formalization scope. Done would require a repository-compatible conjecture statement or a clearly bounded set of prerequisite definitions and proofs.

Written by the indexing model from the issue text.