Loogle!

Result

Found 11 declarations mentioning QuotientGroup.map.

• QuotientGroup.map 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} {H : Type u_2} [Group G] [Group H] (N : Subgroup G) [nN : N.Normal] (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ Subgroup.comap f M) : G ⧸ N →* H ⧸ M
• QuotientGroup.ker_map 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} {H : Type u_2} [Group G] [Group H] (N : Subgroup G) [nN : N.Normal] (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ Subgroup.comap f M) : (QuotientGroup.map N M f h).ker = Subgroup.map (QuotientGroup.mk' N) (Subgroup.comap f M)
• QuotientGroup.map_id 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} [Group G] (N : Subgroup G) [nN : N.Normal] (h : N ≤ Subgroup.comap (MonoidHom.id G) N := ⋯) : QuotientGroup.map N N (MonoidHom.id G) h = MonoidHom.id (G ⧸ N)
• QuotientGroup.map_id_apply 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} [Group G] (N : Subgroup G) [nN : N.Normal] (h : N ≤ Subgroup.comap (MonoidHom.id G) N := ⋯) (x : G ⧸ N) : (QuotientGroup.map N N (MonoidHom.id G) h) x = x
• QuotientGroup.map_mk 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} {H : Type u_2} [Group G] [Group H] (N : Subgroup G) [nN : N.Normal] (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ Subgroup.comap f M) (x : G) : (QuotientGroup.map N M f h) ↑x = ↑(f x)
• QuotientGroup.map_surjective_of_surjective 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} {H : Type u_2} [Group G] [Group H] (N : Subgroup G) [nN : N.Normal] (M : Subgroup H) [M.Normal] (f : G →* H) (hf : Function.Surjective (QuotientGroup.mk ∘ ⇑f)) (h : N ≤ Subgroup.comap f M) : Function.Surjective ⇑(QuotientGroup.map N M f h)
• QuotientGroup.map_mk' 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} {H : Type u_2} [Group G] [Group H] (N : Subgroup G) [nN : N.Normal] (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ Subgroup.comap f M) (x : G) : (QuotientGroup.map N M f h) ((QuotientGroup.mk' N) x) = ↑(f x)
• QuotientGroup.map_comp_map 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} {H : Type u_2} [Group G] [Group H] (N : Subgroup G) [nN : N.Normal] {I : Type u_5} [Group I] (M : Subgroup H) (O : Subgroup I) [M.Normal] [O.Normal] (f : G →* H) (g : H →* I) (hf : N ≤ Subgroup.comap f M) (hg : M ≤ Subgroup.comap g O) (hgf : N ≤ Subgroup.comap (g.comp f) O := ⋯) : (QuotientGroup.map M O g hg).comp (QuotientGroup.map N M f hf) = QuotientGroup.map N O (g.comp f) hgf
• QuotientGroup.map_map 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u_1} {H : Type u_2} [Group G] [Group H] (N : Subgroup G) [nN : N.Normal] {I : Type u_5} [Group I] (M : Subgroup H) (O : Subgroup I) [M.Normal] [O.Normal] (f : G →* H) (g : H →* I) (hf : N ≤ Subgroup.comap f M) (hg : M ≤ Subgroup.comap g O) (hgf : N ≤ Subgroup.comap (g.comp f) O := ⋯) (x : G ⧸ N) : (QuotientGroup.map M O g hg) ((QuotientGroup.map N M f hf) x) = (QuotientGroup.map N O (g.comp f) hgf) x
• QuotientGroup.quotientQuotientEquivQuotientAux_mk 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] (M : Subgroup G) [nM : M.Normal] (h : N ≤ M) (x : G ⧸ N) : (QuotientGroup.quotientQuotientEquivQuotientAux N M h) ↑x = (QuotientGroup.map N M (MonoidHom.id G) h) x
• Ideal.Quotient.stabilizerHomSurjectiveAuxFunctor_aux 📋 Mathlib.RingTheory.Invariant.Profinite
  {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] {G : Type u} [Group G] [MulSemiringAction G B] [SMulCommClass G A B] [TopologicalSpace G] (Q : Ideal B) {N N' : OpenNormalSubgroup G} (e : N ≤ N') (x : G ⧸ ↑N.toOpenSubgroup) (hx : x ∈ MulAction.stabilizer (G ⧸ ↑N.toOpenSubgroup) (Ideal.under (↥(FixedPoints.subalgebra A B ↥↑N.toOpenSubgroup)) Q)) : (QuotientGroup.map (↑N.toOpenSubgroup) N'.toOpenSubgroup.1 (MonoidHom.id G) e) x ∈ MulAction.stabilizer (G ⧸ ↑N'.toOpenSubgroup) (Ideal.under (↥(FixedPoints.subalgebra A B ↥↑N'.toOpenSubgroup)) Q)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 88c39f3 serving mathlib revision 9977002