Finitely generated monoids and groups #

We define finitely generated monoids and groups. See also Submodule.FG and Module.Finite for finitely-generated modules.

Main definition #

Submonoid.FG S, AddSubmonoid.FG S : A submonoid S is finitely generated.
Monoid.FG M, AddMonoid.FG M : A typeclass indicating a type M is finitely generated as a monoid.
Subgroup.FG S, AddSubgroup.FG S : A subgroup S is finitely generated.
Group.FG M, AddGroup.FG M : A typeclass indicating a type M is finitely generated as a group.

Monoids and submonoids #

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def Submonoid.FG {M : Type u_1} [Monoid M] (P : Submonoid M) :

Prop

A submonoid of M is finitely generated if it is the closure of a finite subset of M.

Equations

P.FG = ∃ (S : Finset M), Submonoid.closure ↑S = P

Instances For

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def AddSubmonoid.FG {M : Type u_1} [AddMonoid M] (P : AddSubmonoid M) :

Prop

An additive submonoid of N is finitely generated if it is the closure of a finite subset of M.

Equations

P.FG = ∃ (S : Finset M), AddSubmonoid.closure ↑S = P

Instances For

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theorem Submonoid.fg_iff {M : Type u_1} [Monoid M] (P : Submonoid M) :

P.FG ↔ ∃ (S : Set M), closure S = P ∧ S.Finite

An equivalent expression of Submonoid.FG in terms of Set.Finite instead of Finset.

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theorem AddSubmonoid.fg_iff {M : Type u_1} [AddMonoid M] (P : AddSubmonoid M) :

P.FG ↔ ∃ (S : Set M), closure S = P ∧ S.Finite

An equivalent expression of AddSubmonoid.FG in terms of Set.Finite instead of Finset.

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theorem Submonoid.FG.exists_minimal_closure_eq {M : Type u_1} [Monoid M] {P : Submonoid M} (hP : P.FG) :

∃ (S : Finset M), Minimal (fun (x : Finset M) => closure ↑x = P) S

A finitely generated submonoid has a minimal generating set.

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theorem AddSubmonoid.FG.exists_minimal_closure_eq {M : Type u_1} [AddMonoid M] {P : AddSubmonoid M} (hP : P.FG) :

∃ (S : Finset M), Minimal (fun (x : Finset M) => closure ↑x = P) S

A finitely generated submonoid has a minimal generating set.

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theorem Submonoid.fg_iff_add_fg {M : Type u_1} [Monoid M] (P : Submonoid M) :

P.FG ↔ (toAddSubmonoid P).FG

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theorem AddSubmonoid.fg_iff_mul_fg {M : Type u_3} [AddMonoid M] (P : AddSubmonoid M) :

P.FG ↔ (toSubmonoid P).FG

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theorem Submonoid.FG.prod {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] {P : Submonoid M} {Q : Submonoid N} (hP : P.FG) (hQ : Q.FG) :

(P.prod Q).FG

The product of two finitely generated submonoids is finitely generated.

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theorem AddSubmonoid.FG.prod {M : Type u_1} {N : Type u_2} [AddMonoid M] [AddMonoid N] {P : AddSubmonoid M} {Q : AddSubmonoid N} (hP : P.FG) (hQ : Q.FG) :

(P.prod Q).FG

The product of two finitely generated additive submonoids is finitely generated.

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@[deprecated AddSubmonoid.FG.prod (since := "2025-08-28")]

theorem AddSubmonoid.FG.sum {M : Type u_1} {N : Type u_2} [AddMonoid M] [AddMonoid N] {P : AddSubmonoid M} {Q : AddSubmonoid N} (hP : P.FG) (hQ : Q.FG) :

(P.prod Q).FG

Alias of AddSubmonoid.FG.prod.

The product of two finitely generated additive submonoids is finitely generated.

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theorem Submonoid.iSup_map_mulSingle {ι : Type u_3} [Finite ι] {M : ι → Type u_4} [(i : ι) → Monoid (M i)] {P : (i : ι) → Submonoid (M i)} [DecidableEq ι] :

⨆ (i : ι), map (MonoidHom.mulSingle M i) (P i) = pi Set.univ P

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theorem AddSubmonoid.iSup_map_single {ι : Type u_3} [Finite ι] {M : ι → Type u_4} [(i : ι) → AddMonoid (M i)] {P : (i : ι) → AddSubmonoid (M i)} [DecidableEq ι] :

⨆ (i : ι), map (AddMonoidHom.single M i) (P i) = pi Set.univ P

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theorem Submonoid.FG.pi {ι : Type u_3} [Finite ι] {M : ι → Type u_4} [(i : ι) → Monoid (M i)] {P : (i : ι) → Submonoid (M i)} (hP : ∀ (i : ι), (P i).FG) :

(Submonoid.pi Set.univ P).FG

Finite product of finitely generated submonoids is finitely generated.

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theorem AddSubmonoid.FG.pi {ι : Type u_3} [Finite ι] {M : ι → Type u_4} [(i : ι) → AddMonoid (M i)] {P : (i : ι) → AddSubmonoid (M i)} (hP : ∀ (i : ι), (P i).FG) :

(AddSubmonoid.pi Set.univ P).FG

Finite product of finitely generated additive submonoids is finitely generated.

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class AddMonoid.FG (M : Type u_3) [AddMonoid M] :

Prop

An additive monoid is finitely generated if it is finitely generated as an additive submonoid of itself.

fg_top : ⊤.FG

Instances

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theorem AddMonoid.fG_iff (M : Type u_3) [AddMonoid M] :

FG M ↔ ⊤.FG

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class Monoid.FG (M : Type u_1) [Monoid M] :

Prop

A monoid is finitely generated if it is finitely generated as a submonoid of itself.

fg_top : ⊤.FG

Instances

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theorem Monoid.fg_def {M : Type u_1} [Monoid M] :

FG M ↔ ⊤.FG

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theorem AddMonoid.fg_def {M : Type u_1} [AddMonoid M] :

FG M ↔ ⊤.FG

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theorem Monoid.fg_iff {M : Type u_1} [Monoid M] :

FG M ↔ ∃ (S : Set M), Submonoid.closure S = ⊤ ∧ S.Finite

An equivalent expression of Monoid.FG in terms of Set.Finite instead of Finset.

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theorem AddMonoid.fg_iff {M : Type u_1} [AddMonoid M] :

FG M ↔ ∃ (S : Set M), AddSubmonoid.closure S = ⊤ ∧ S.Finite

An equivalent expression of AddMonoid.FG in terms of Set.Finite instead of Finset.

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theorem Submonoid.exists_minimal_closure_eq_top (M : Type u_1) [Monoid M] [Monoid.FG M] :

∃ (S : Finset M), Minimal (fun (x : Finset M) => closure ↑x = ⊤) S

A finitely generated monoid has a minimal generating set.

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theorem AddSubmonoid.exists_minimal_closure_eq_top (M : Type u_1) [AddMonoid M] [AddMonoid.FG M] :

∃ (S : Finset M), Minimal (fun (x : Finset M) => closure ↑x = ⊤) S

A finitely generated monoid has a minimal generating set.

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theorem Monoid.fg_iff_add_fg {M : Type u_1} [Monoid M] :

FG M ↔ AddMonoid.FG (Additive M)

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theorem AddMonoid.fg_iff_mul_fg {M : Type u_3} [AddMonoid M] :

FG M ↔ Monoid.FG (Multiplicative M)

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instance AddMonoid.fg_of_monoid_fg {M : Type u_1} [Monoid M] [Monoid.FG M] :

FG (Additive M)

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instance Monoid.fg_of_addMonoid_fg {M : Type u_3} [AddMonoid M] [AddMonoid.FG M] :

FG (Multiplicative M)

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theorem Monoid.fg_of_finite {M : Type u_1} [Monoid M] [Finite M] :

FG M

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theorem AddMonoid.fg_of_finite {M : Type u_1} [AddMonoid M] [Finite M] :

FG M

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theorem Submonoid.FG.map {M : Type u_1} [Monoid M] {M' : Type u_3} [Monoid M'] {P : Submonoid M} (h : P.FG) (e : M →* M') :

(Submonoid.map e P).FG

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theorem AddSubmonoid.FG.map {M : Type u_1} [AddMonoid M] {M' : Type u_3} [AddMonoid M'] {P : AddSubmonoid M} (h : P.FG) (e : M →+ M') :

(AddSubmonoid.map e P).FG

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theorem Submonoid.FG.map_injective {M : Type u_1} [Monoid M] {M' : Type u_3} [Monoid M'] {P : Submonoid M} (e : M →* M') (he : Function.Injective ⇑e) (h : (Submonoid.map e P).FG) :

P.FG

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theorem AddSubmonoid.FG.map_injective {M : Type u_1} [AddMonoid M] {M' : Type u_3} [AddMonoid M'] {P : AddSubmonoid M} (e : M →+ M') (he : Function.Injective ⇑e) (h : (AddSubmonoid.map e P).FG) :

P.FG

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@[simp]

theorem Monoid.fg_iff_submonoid_fg {M : Type u_1} [Monoid M] (N : Submonoid M) :

FG ↥N ↔ N.FG

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@[simp]

theorem AddMonoid.fg_iff_addSubmonoid_fg {M : Type u_1} [AddMonoid M] (N : AddSubmonoid M) :

FG ↥N ↔ N.FG

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theorem Monoid.fg_of_surjective {M : Type u_1} [Monoid M] {M' : Type u_3} [Monoid M'] [FG M] (f : M →* M') (hf : Function.Surjective ⇑f) :

FG M'

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theorem AddMonoid.fg_of_surjective {M : Type u_1} [AddMonoid M] {M' : Type u_3} [AddMonoid M'] [FG M] (f : M →+ M') (hf : Function.Surjective ⇑f) :

FG M'

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instance Monoid.fg_range {M : Type u_1} [Monoid M] {M' : Type u_3} [Monoid M'] [FG M] (f : M →* M') :

FG ↥(MonoidHom.mrange f)

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instance AddMonoid.fg_range {M : Type u_1} [AddMonoid M] {M' : Type u_3} [AddMonoid M'] [FG M] (f : M →+ M') :

FG ↥(AddMonoidHom.mrange f)

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theorem Submonoid.powers_fg {M : Type u_1} [Monoid M] (r : M) :

(powers r).FG

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theorem AddSubmonoid.multiples_fg {M : Type u_1} [AddMonoid M] (r : M) :

(multiples r).FG

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instance Monoid.powers_fg {M : Type u_1} [Monoid M] (r : M) :

FG ↥(Submonoid.powers r)

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instance AddMonoid.multiples_fg {M : Type u_1} [AddMonoid M] (r : M) :

FG ↥(AddSubmonoid.multiples r)

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instance Monoid.closure_finset_fg {M : Type u_1} [Monoid M] (s : Finset M) :

FG ↥(Submonoid.closure ↑s)

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instance AddMonoid.closure_finset_fg {M : Type u_1} [AddMonoid M] (s : Finset M) :

FG ↥(AddSubmonoid.closure ↑s)

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instance Monoid.closure_finite_fg {M : Type u_1} [Monoid M] (s : Set M) [Finite ↑s] :

FG ↥(Submonoid.closure s)

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instance AddMonoid.closure_finite_fg {M : Type u_1} [AddMonoid M] (s : Set M) [Finite ↑s] :

FG ↥(AddSubmonoid.closure s)

Groups and subgroups #

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def Subgroup.FG {G : Type u_3} [Group G] (P : Subgroup G) :

Prop

A subgroup of G is finitely generated if it is the closure of a finite subset of G.

Equations

P.FG = ∃ (S : Finset G), Subgroup.closure ↑S = P

Instances For

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def AddSubgroup.FG {G : Type u_3} [AddGroup G] (P : AddSubgroup G) :

Prop

An additive subgroup of H is finitely generated if it is the closure of a finite subset of H.

Equations

P.FG = ∃ (S : Finset G), AddSubgroup.closure ↑S = P

Instances For

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theorem Subgroup.fg_iff {G : Type u_3} [Group G] (P : Subgroup G) :

P.FG ↔ ∃ (S : Set G), closure S = P ∧ S.Finite

An equivalent expression of Subgroup.FG in terms of Set.Finite instead of Finset.

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theorem AddSubgroup.fg_iff {G : Type u_3} [AddGroup G] (P : AddSubgroup G) :

P.FG ↔ ∃ (S : Set G), closure S = P ∧ S.Finite

An equivalent expression of AddSubgroup.fg in terms of Set.Finite instead of Finset.

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theorem Subgroup.fg_iff_submonoid_fg {G : Type u_3} [Group G] (P : Subgroup G) :

P.FG ↔ P.FG

A subgroup is finitely generated if and only if it is finitely generated as a submonoid.

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theorem AddSubgroup.fg_iff_addSubmonoid_fg {G : Type u_3} [AddGroup G] (P : AddSubgroup G) :

P.FG ↔ P.FG

An additive subgroup is finitely generated if and only if it is finitely generated as an additive submonoid.

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theorem Subgroup.fg_iff_add_fg {G : Type u_3} [Group G] (P : Subgroup G) :

P.FG ↔ (toAddSubgroup P).FG

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theorem AddSubgroup.fg_iff_mul_fg {H : Type u_4} [AddGroup H] (P : AddSubgroup H) :

P.FG ↔ (toSubgroup P).FG

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theorem Subgroup.FG.prod {G : Type u_3} [Group G] {G' : Type u_5} [Group G'] {P : Subgroup G} {Q : Subgroup G'} (hP : P.FG) (hQ : Q.FG) :

(P.prod Q).FG

The product of two finitely generated subgroups is finitely generated.

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theorem AddSubgroup.FG.prod {G : Type u_3} [AddGroup G] {G' : Type u_5} [AddGroup G'] {P : AddSubgroup G} {Q : AddSubgroup G'} (hP : P.FG) (hQ : Q.FG) :

(P.prod Q).FG

The product of two finitely generated additive subgroups is finitely generated.

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theorem Subgroup.FG.pi {ι : Type u_5} [Finite ι] {G : ι → Type u_6} [(i : ι) → Group (G i)] {P : (i : ι) → Subgroup (G i)} (hP : ∀ (i : ι), (P i).FG) :

(Subgroup.pi Set.univ P).FG

Finite product of finitely generated subgroups is finitely generated.

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theorem AddSubgroup.FG.pi {ι : Type u_5} [Finite ι] {G : ι → Type u_6} [(i : ι) → AddGroup (G i)] {P : (i : ι) → AddSubgroup (G i)} (hP : ∀ (i : ι), (P i).FG) :

(AddSubgroup.pi Set.univ P).FG

Finite product of finitely generated additive subgroups is finitely generated.

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class Group.FG (G : Type u_3) [Group G] :

Prop

A group is finitely generated if it is finitely generated as a subgroup of itself.

out : ⊤.FG

Instances

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class AddGroup.FG (H : Type u_4) [AddGroup H] :

Prop

An additive group is finitely generated if it is finitely generated as an additive subgroup of itself.

out : ⊤.FG

Instances

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theorem Group.fg_def {G : Type u_3} [Group G] :

FG G ↔ ⊤.FG

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theorem AddGroup.fg_def {H : Type u_4} [AddGroup H] :

FG H ↔ ⊤.FG

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theorem Group.fg_iff {G : Type u_3} [Group G] :

FG G ↔ ∃ (S : Set G), Subgroup.closure S = ⊤ ∧ S.Finite

An equivalent expression of Group.FG in terms of Set.Finite instead of Finset.

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theorem AddGroup.fg_iff {G : Type u_3} [AddGroup G] :

FG G ↔ ∃ (S : Set G), AddSubgroup.closure S = ⊤ ∧ S.Finite

An equivalent expression of AddGroup.fg in terms of Set.Finite instead of Finset.

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theorem Group.fg_iff' {G : Type u_3} [Group G] :

FG G ↔ ∃ (n : ℕ) (S : Finset G), S.card = n ∧ Subgroup.closure ↑S = ⊤

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theorem AddGroup.fg_iff' {G : Type u_3} [AddGroup G] :

FG G ↔ ∃ (n : ℕ) (S : Finset G), S.card = n ∧ AddSubgroup.closure ↑S = ⊤

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theorem Group.fg_iff_monoid_fg {G : Type u_3} [Group G] :

FG G ↔ Monoid.FG G

A group is finitely generated if and only if it is finitely generated as a monoid.

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theorem AddGroup.fg_iff_addMonoid_fg {G : Type u_3} [AddGroup G] :

FG G ↔ AddMonoid.FG G

An additive group is finitely generated if and only if it is finitely generated as an additive monoid.

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instance Monoid.fg_of_group_fg {G : Type u_3} [Group G] [Group.FG G] :

FG G

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instance AddMonoid.fg_of_addGroup_fg {G : Type u_3} [AddGroup G] [AddGroup.FG G] :

FG G

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@[simp]

theorem Group.fg_iff_subgroup_fg {G : Type u_3} [Group G] (H : Subgroup G) :

FG ↥H ↔ H.FG

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@[simp]

theorem AddGroup.fg_iff_addSubgroup_fg {G : Type u_3} [AddGroup G] (H : AddSubgroup G) :

FG ↥H ↔ H.FG

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theorem GroupFG.iff_add_fg {G : Type u_3} [Group G] :

Group.FG G ↔ AddGroup.FG (Additive G)

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theorem AddGroup.fg_iff_mul_fg {H : Type u_4} [AddGroup H] :

FG H ↔ Group.FG (Multiplicative H)

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instance AddGroup.fg_of_group_fg {G : Type u_3} [Group G] [Group.FG G] :

FG (Additive G)

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instance Group.fg_of_mul_group_fg {H : Type u_4} [AddGroup H] [AddGroup.FG H] :

FG (Multiplicative H)

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@[instance 100]

instance Group.fg_of_finite {G : Type u_3} [Group G] [Finite G] :

FG G

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@[instance 100]

instance AddGroup.fg_of_finite {G : Type u_3} [AddGroup G] [Finite G] :

FG G

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theorem Group.fg_of_surjective {G : Type u_3} [Group G] {G' : Type u_5} [Group G'] [hG : FG G] {f : G →* G'} (hf : Function.Surjective ⇑f) :

FG G'

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theorem AddGroup.fg_of_surjective {G : Type u_3} [AddGroup G] {G' : Type u_5} [AddGroup G'] [hG : FG G] {f : G →+ G'} (hf : Function.Surjective ⇑f) :

FG G'

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instance Group.fg_range {G : Type u_3} [Group G] {G' : Type u_5} [Group G'] [FG G] (f : G →* G') :

FG ↥f.range

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instance AddGroup.fg_range {G : Type u_3} [AddGroup G] {G' : Type u_5} [AddGroup G'] [FG G] (f : G →+ G') :

FG ↥f.range

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instance Group.closure_finset_fg {G : Type u_3} [Group G] (s : Finset G) :

FG ↥(Subgroup.closure ↑s)

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instance AddGroup.closure_finset_fg {G : Type u_3} [AddGroup G] (s : Finset G) :

FG ↥(AddSubgroup.closure ↑s)

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instance Group.closure_finite_fg {G : Type u_3} [Group G] (s : Set G) [Finite ↑s] :

FG ↥(Subgroup.closure s)

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instance AddGroup.closure_finite_fg {G : Type u_3} [AddGroup G] (s : Set G) [Finite ↑s] :

FG ↥(AddSubgroup.closure s)

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instance QuotientGroup.fg {G : Type u_3} [Group G] [Group.FG G] (N : Subgroup G) [N.Normal] :

Group.FG (G ⧸ N)

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instance QuotientAddGroup.fg {G : Type u_3} [AddGroup G] [AddGroup.FG G] (N : AddSubgroup G) [N.Normal] :

AddGroup.FG (G ⧸ N)

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instance Prod.instMonoidFG {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] [Monoid.FG M] [Monoid.FG N] :

Monoid.FG (M × N)

The product of two finitely generated monoids is finitely generated.

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instance Prod.instAddMonoidFG {M : Type u_1} {N : Type u_2} [AddMonoid M] [AddMonoid N] [AddMonoid.FG M] [AddMonoid.FG N] :

AddMonoid.FG (M × N)

The product of two finitely generated additive monoids is finitely generated.

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instance Prod.instGroupFG {G : Type u_3} [Group G] {G' : Type u_5} [Group G'] [Group.FG G] [Group.FG G'] :

Group.FG (G × G')

The product of two finitely generated groups is finitely generated.

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instance Prod.instAddGroupFG {G : Type u_3} [AddGroup G] {G' : Type u_5} [AddGroup G'] [AddGroup.FG G] [AddGroup.FG G'] :

AddGroup.FG (G × G')

The product of two finitely generated additive groups is finitely generated.

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instance Pi.instMonoidFG {ι : Type u_5} [Finite ι] {M : ι → Type u_6} [(i : ι) → Monoid (M i)] [∀ (i : ι), Monoid.FG (M i)] :

Monoid.FG ((i : ι) → M i)

Finite product of finitely generated monoids is finitely generated.

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instance Pi.instAddMonoidFG {ι : Type u_5} [Finite ι] {M : ι → Type u_6} [(i : ι) → AddMonoid (M i)] [∀ (i : ι), AddMonoid.FG (M i)] :

AddMonoid.FG ((i : ι) → M i)

Finite product of finitely generated additive monoids is finitely generated.

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instance Pi.instGroupFG {ι : Type u_5} [Finite ι] {G : ι → Type u_6} [(i : ι) → Group (G i)] [∀ (i : ι), Group.FG (G i)] :

Group.FG ((i : ι) → G i)

Finite product of finitely generated groups is finitely generated.

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instance Pi.instAddGroupFG {ι : Type u_5} [Finite ι] {G : ι → Type u_6} [(i : ι) → AddGroup (G i)] [∀ (i : ι), AddGroup.FG (G i)] :

AddGroup.FG ((i : ι) → G i)

Finite product of finitely generated additive groups is finitely generated.

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instance AddMonoid.instFGNat :

FG ℕ

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instance AddGroup.instFGInt :

FG ℤ

Documentation

Mathlib.GroupTheory.Finiteness

Finitely generated monoids and groups #

Main definition #

Monoids and submonoids #

Groups and subgroups #