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

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

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

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.

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.

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

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

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 finitely generated submonoids is finitely generated.

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

The product of finitely generated submonoids is finitely generated.

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

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

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

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

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

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.

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.

theorem Monoid.fg_iff_add_fg {M : Type u_1} [Monoid M] : FG M ↔ AddMonoid.FG (Additive M)

theorem AddMonoid.fg_iff_mul_fg {M : Type u_3} [AddMonoid M] : FG M ↔ Monoid.FG (Multiplicative M)

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

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

@[instance 100]

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

@[instance 100]

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

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

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

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

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

@[simp]

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

@[simp]

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

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'

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'

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)

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)

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

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

instance Monoid.powers_fg {M : Type u_1} [Monoid M] (r : M) : FG ↥(Submonoid.powers r)

instance AddMonoid.multiples_fg {M : Type u_1} [AddMonoid M] (r : M) : FG ↥(AddSubmonoid.multiples r)

instance Monoid.closure_finset_fg {M : Type u_1} [Monoid M] (s : Finset M) : FG ↥(Submonoid.closure ↑s)

instance AddMonoid.closure_finset_fg {M : Type u_1} [AddMonoid M] (s : Finset M) : FG ↥(AddSubmonoid.closure ↑s)

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

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

Groups and subgroups

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

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

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.

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.

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.

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.

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

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

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

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

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

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

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.

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.

theorem Group.fg_iff' {G : Type u_3} [Group G] : FG G ↔ ∃ (n : ℕ) (S : Finset G), S.card = n ∧ Subgroup.closure ↑S = ⊤

theorem AddGroup.fg_iff' {G : Type u_3} [AddGroup G] : FG G ↔ ∃ (n : ℕ) (S : Finset G), S.card = n ∧ AddSubgroup.closure ↑S = ⊤

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.

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.

@[simp]

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

@[simp]

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

theorem GroupFG.iff_add_fg {G : Type u_3} [Group G] : Group.FG G ↔ AddGroup.FG (Additive G)

theorem AddGroup.fg_iff_mul_fg {H : Type u_4} [AddGroup H] : FG H ↔ Group.FG (Multiplicative H)

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

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

@[instance 100]

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

@[instance 100]

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

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'

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'

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

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

instance Group.closure_finset_fg {G : Type u_3} [Group G] (s : Finset G) : FG ↥(Subgroup.closure ↑s)

instance AddGroup.closure_finset_fg {G : Type u_3} [AddGroup G] (s : Finset G) : FG ↥(AddSubgroup.closure ↑s)

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

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

noncomputable def Group.rank (G : Type u_3) [Group G] [h : FG G] : ℕ

The minimum number of generators of a group.

Equations

• Group.rank G = Nat.find ⋯

noncomputable def AddGroup.rank (G : Type u_3) [AddGroup G] [h : FG G] : ℕ

The minimum number of generators of an additive group

Equations

• AddGroup.rank G = Nat.find ⋯

theorem Group.rank_spec (G : Type u_3) [Group G] [h : FG G] : ∃ (S : Finset G), S.card = rank G ∧ Subgroup.closure ↑S = ⊤

theorem AddGroup.rank_spec (G : Type u_3) [AddGroup G] [h : FG G] : ∃ (S : Finset G), S.card = rank G ∧ AddSubgroup.closure ↑S = ⊤

theorem Group.rank_le (G : Type u_3) [Group G] [h : FG G] {S : Finset G} (hS : Subgroup.closure ↑S = ⊤) : rank G ≤ S.card

theorem AddGroup.rank_le (G : Type u_3) [AddGroup G] [h : FG G] {S : Finset G} (hS : AddSubgroup.closure ↑S = ⊤) : rank G ≤ S.card

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

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

theorem Group.rank_range_le {G : Type u_3} [Group G] {G' : Type u_5} [Group G'] [FG G] {f : G →* G'} : rank ↥f.range ≤ rank G

theorem AddGroup.rank_range_le {G : Type u_3} [AddGroup G] {G' : Type u_5} [AddGroup G'] [FG G] {f : G →+ G'} : rank ↥f.range ≤ rank G

theorem Group.rank_congr {G : Type u_3} [Group G] {G' : Type u_5} [Group G'] [FG G] [FG G'] (f : G ≃* G') : rank G = rank G'

theorem AddGroup.rank_congr {G : Type u_3} [AddGroup G] {G' : Type u_5} [AddGroup G'] [FG G] [FG G'] (f : G ≃+ G') : rank G = rank G'

theorem Subgroup.rank_congr {G : Type u_3} [Group G] {H K : Subgroup G} [Group.FG ↥H] [Group.FG ↥K] (h : H = K) : Group.rank ↥H = Group.rank ↥K

theorem AddSubgroup.rank_congr {G : Type u_3} [AddGroup G] {H K : AddSubgroup G} [AddGroup.FG ↥H] [AddGroup.FG ↥K] (h : H = K) : AddGroup.rank ↥H = AddGroup.rank ↥K

theorem Subgroup.rank_closure_finset_le_card {G : Type u_3} [Group G] (s : Finset G) : Group.rank ↥(closure ↑s) ≤ s.card

theorem AddSubgroup.rank_closure_finset_le_card {G : Type u_3} [AddGroup G] (s : Finset G) : AddGroup.rank ↥(closure ↑s) ≤ s.card

theorem Subgroup.rank_closure_finite_le_nat_card {G : Type u_3} [Group G] (s : Set G) [Finite ↑s] : Group.rank ↥(closure s) ≤ Nat.card ↑s

theorem AddSubgroup.rank_closure_finite_le_nat_card {G : Type u_3} [AddGroup G] (s : Set G) [Finite ↑s] : AddGroup.rank ↥(closure s) ≤ Nat.card ↑s

theorem Subgroup.nat_card_centralizer_nat_card_stabilizer {G : Type u_3} [Group G] (g : G) : Nat.card ↥(centralizer {g}) = Nat.card ↥(MulAction.stabilizer (ConjAct G) g)

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

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

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 finitely generated monoids is finitely generated.

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 finitely generated monoids is finitely generated.