Subgroup_Product_is_Internal_Direct_Product_iff_Surjective

⚠️ 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.

The product of subgroups H and K generates G (H ∨ K = G) if and only if the multiplication map H × K → G is surjective.

Why this node was retired

The posted statement is

import Mathlib.Analysis.Complex.Basic

theorem Subgroup_Product_is_Internal_Direct_Product_iff_Surjective {G : Type _} [Group G] (H K : Subgroup G) : (H ⊔ K = ⊤) ↔ ∀ g : G, ∃ (h : H) (k : K), (h : G) * (k : G) = g := by sorry

The subgroup join is substituted for the setwise product.

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

Proposed corrected statement

For subgroups H,K, the multiplication map H×K→G is surjective iff the setwise product HK=G. If retaining H⊔K=⊤, additionally require H and K to permute (HK=KH), for example by requiring one subgroup normal. For an internal direct product also require trivial intersection and cross commutation.

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.