Second_Isomorphism_Theorem_v3

⚠️ 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 second isomorphism theorem: For a subgroup H and normal subgroup N, the natural map H -> G/N has kernel H ∩ N and image HN/N.

Why this node was retired

The posted statement is

import Mathlib.Analysis.Complex.Basic
import Mathlib.GroupTheory.QuotientGroup.Basic

theorem Second_Isomorphism_Theorem_v3 {G : Type _} [Group G] (H N : Subgroup G) [hN : N.Normal] : ∃ f : H →* G ⧸ N, Function.Surjective f ∧ f.ker = N.subgroupOf H := by sorry

Surjectivity onto all G/N replaces surjectivity onto the image HN/N.

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

Proposed corrected statement

The natural map H→G/N has kernel H∩N and image HN/N; assert H/(H∩N)≃HN/N. Alternatively keep surjectivity onto G/N and add H⊔N=⊤ with N normal.

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.