Isomorphism_of_Abelian_Groups
Disproved⚠️ Retired — specification defect
The Lean statement below does not encode the problem shown on this page, so its
Disprovedstatus carries no information about that problem. Do not import this node or use it as a dependency.
A group isomorphic to an abelian group is also abelian.
Why this node was retired
The posted statement is
theorem Isomorphism_of_Abelian_Groups {G H : Type _} [Group G] [Group H] (f : G ≃* H) [CommGroup G] : ∀ a b : H, a * b = b * a := by sorry
The isomorphism uses an earlier Group G, while a later independent CommGroup G may have different multiplication.
The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.
Proposed corrected statement
Declare [CommGroup G] [Group H] before f:G≃*H, using only that domain group structure; then H is abelian.
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.
import Mathlib.Algebra.Group.Equiv.Basic
theorem Isomorphism_of_Abelian_Groups {G H : Type _} [Group G] [Group H] (f : G ≃* H) [CommGroup G] : ∀ a b : H, a * b = b * a := by sorry