Theorems · Theorem · group theory
Subgroup.exists_mulEquiv_eq_graph
∀ {H : Type u_2} {I : Type u_3} [inst : Group H] [inst_1 : Group I] {G : Subgroup (H × I)},
Function.Bijective (Prod.fst ∘ ⇑G.subtype) → Function.Bijective (Prod.snd ∘ ⇑G.subtype) → ∃ e, G = e.toMonoidHom.graphGoursat's lemma for monoid isomorphisms.
Let G ≤ H × I be a submonoid of a product of monoids. Assume that the natural maps from G to
both factors are bijective. Then G is the graph of some isomorphism f : H ≃* I.
- Defined in
- Mathlib.Algebra.Group.Graph
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 24 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Groupstatement and proof · cited by 6,238
- MonoidHomstatement · cited by 3,629
- Subgroupstatement and proof · cited by 3,593
- MulEquivstatement and proof · cited by 1,142
- Function.Bijectivestatement and proof · cited by 863
- Subgroup.subtypestatement and proof · cited by 185
- MulEquiv.toMonoidHomstatement · cited by 126
- MonoidHom.graphstatement · cited by 10
- Submonoid.exists_mulEquiv_eq_mgraphproof · cited by 1
Cited by0
Results whose statement or proof uses this declaration.
Nothing cites this yet.