Loogle!

Result

Found 8 definitions mentioning MonoidHom and IsCyclic.

• commGroupOfCyclicCenterQuotient 📋 Mathlib.GroupTheory.SpecificGroups.Cyclic
  {G : Type u_2} {G' : Type u_3} [Group G] [Group G'] [IsCyclic G'] (f : G →* G') (hf : f.ker ≤ Subgroup.center G) : CommGroup G
• MonoidHom.map_cyclic 📋 Mathlib.GroupTheory.SpecificGroups.Cyclic
  {G : Type u_2} [Group G] [h : IsCyclic G] (σ : G →* G) : ∃ m, ∀ (g : G), σ g = g ^ m
• commutative_of_cyclic_center_quotient 📋 Mathlib.GroupTheory.SpecificGroups.Cyclic
  {G : Type u_2} {G' : Type u_3} [Group G] [Group G'] [IsCyclic G'] (f : G →* G') (hf : f.ker ≤ Subgroup.center G) (a b : G) : a * b = b * a
• commGroupOfCyclicCenterQuotient.eq_1 📋 Mathlib.GroupTheory.SpecificGroups.Cyclic
  {G : Type u_2} {G' : Type u_3} [Group G] [Group G'] [IsCyclic G'] (f : G →* G') (hf : f.ker ≤ Subgroup.center G) : commGroupOfCyclicCenterQuotient f hf = CommGroup.mk ⋯
• isCyclic_of_subgroup_isDomain 📋 Mathlib.RingTheory.IntegralDomain
  {R : Type u_1} {G : Type u_2} [CommRing R] [IsDomain R] [Group G] [Finite G] (f : G →* R) (hf : Function.Injective ⇑f) : IsCyclic G
• IsCyclic.monoidHom_mulEquiv_rootsOfUnity 📋 Mathlib.RingTheory.RootsOfUnity.Basic
  (G : Type u_7) [CommGroup G] [IsCyclic G] (G' : Type u_8) [CommGroup G'] : Nonempty ((G →* G') ≃* ↥(rootsOfUnity (Nat.card G) G'))
• IsCyclic.exists_apply_ne_one 📋 Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
  {G : Type u_7} {G' : Type u_8} [CommGroup G] [IsCyclic G] [Finite G] [CommGroup G'] (hG' : ∃ ζ, IsPrimitiveRoot ζ (Nat.card G)) ⦃a : G⦄ (ha : a ≠ 1) : ∃ φ, φ a ≠ 1
• IsCyclic.monoidHom_equiv_self 📋 Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity
  (G : Type u_1) (M : Type u_2) [CommGroup G] [Finite G] [IsCyclic G] [CommMonoid M] [HasEnoughRootsOfUnity M (Nat.card G)] : Nonempty ((G →* Mˣ) ≃* G)

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.

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, _ * _, _ ^ _, |- _ < _ → _
woould 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.

This is Loogle revision 4e1aab0 serving mathlib revision 2d53f5f