Internal_Direct_Product_Generated_by_Subgroups

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $G$ be a group whose identity is $e$. Let $\sequence {H_n}$ be a sequence of subgroups of $G$. Then the subgroup generated by $\displaystyle \bigcup_{k \mathop = 1}^n H_k$ is the internal group direct product of $\sequence {H_n}$ $\sequence {H_n}$ is an independent sequence of subgroups such that every element of $H_i$ commutes with every element of $H_j$ whenever $1 \le i < j \le n$.

Why this node was retired

The posted statement is

theorem Internal_Direct_Product_Generated_by_Subgroups {G : Type _} [Group G] (H K : Subgroup G) (hHK : H ⊓ K = ⊥) (hGen : H ⊔ K = ⊤) : ∀ g : G, ∃ h k : G, h ∈ H ∧ k ∈ K ∧ g = h * k := by sorry

Trivial intersection and generation alone do not encode an internal direct product; cross commutation or appropriate normality is missing.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

If H⊓K=⊥, H⊔K=⊤ and every element of H commutes with every element of K, then every g∈G is uniquely h*k with h∈H,k∈K, and H×K≃*G.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.