Deducing finiteness of a group.

@[instance_reducible]

noncomputable def Group.fintypeOfKerLeRange {F : Type u_1} {G : Type u_2} {H : Type u_3} [Group F] [Group G] [Group H] [Fintype F] [Fintype H] (f : F →* G) (g : G →* H) (h : g.ker ≤ f.range) : Fintype G

If F and H are finite such that ker(G →* H) ≤ im(F →* G), then G is finite.

Equations

• Group.fintypeOfKerLeRange f g h = Fintype.ofEquiv ((G ⧸ g.ker) × ↥g.ker) Subgroup.groupEquivQuotientProdSubgroup.symm

@[instance_reducible]

noncomputable def AddGroup.fintypeOfKerLeRange {F : Type u_1} {G : Type u_2} {H : Type u_3} [AddGroup F] [AddGroup G] [AddGroup H] [Fintype F] [Fintype H] (f : F →+ G) (g : G →+ H) (h : g.ker ≤ f.range) : Fintype G

If F and H are finite such that ker(G →+ H) ≤ im(F →+ G), then G is finite.

Equations

• AddGroup.fintypeOfKerLeRange f g h = Fintype.ofEquiv ((G ⧸ g.ker) × ↥g.ker) AddSubgroup.addGroupEquivQuotientProdAddSubgroup.symm

@[instance_reducible]

noncomputable def Group.fintypeOfKerEqRange {F : Type u_1} {G : Type u_2} {H : Type u_3} [Group F] [Group G] [Group H] [Fintype F] [Fintype H] (f : F →* G) (g : G →* H) (h : g.ker = f.range) : Fintype G

If F and H are finite such that ker(G →* H) = im(F →* G), then G is finite.

Equations

• Group.fintypeOfKerEqRange f g h = Group.fintypeOfKerLeRange f g ⋯

@[instance_reducible]

noncomputable def AddGroup.fintypeOfKerEqRange {F : Type u_1} {G : Type u_2} {H : Type u_3} [AddGroup F] [AddGroup G] [AddGroup H] [Fintype F] [Fintype H] (f : F →+ G) (g : G →+ H) (h : g.ker = f.range) : Fintype G

If F and H are finite such that ker(G →+ H) = im(F →+ G), then G is finite.

Equations

• AddGroup.fintypeOfKerEqRange f g h = AddGroup.fintypeOfKerLeRange f g ⋯

@[instance_reducible]

noncomputable def Group.fintypeOfKerOfCodom {G : Type u_2} {H : Type u_3} [Group G] [Group H] [Fintype H] (g : G →* H) [Fintype ↥g.ker] : Fintype G

If ker(G →* H) and H are finite, then G is finite.

Equations

• Group.fintypeOfKerOfCodom g = Group.fintypeOfKerLeRange (Subgroup.topEquiv.toMonoidHom.comp (Subgroup.inclusion ⋯)) g ⋯

@[instance_reducible]

noncomputable def AddGroup.fintypeOfKerOfCodom {G : Type u_2} {H : Type u_3} [AddGroup G] [AddGroup H] [Fintype H] (g : G →+ H) [Fintype ↥g.ker] : Fintype G

If ker(G →+ H) and H are finite, then G is finite.

Equations

• AddGroup.fintypeOfKerOfCodom g = AddGroup.fintypeOfKerLeRange (AddSubgroup.topEquiv.toAddMonoidHom.comp (AddSubgroup.inclusion ⋯)) g ⋯

@[instance_reducible]

noncomputable def Group.fintypeOfDomOfCoker {F : Type u_1} {G : Type u_2} [Group F] [Group G] [Fintype F] (f : F →* G) [f.range.Normal] [Fintype (G ⧸ f.range)] : Fintype G

If F and coker(F →* G) are finite, then G is finite.

Equations

• Group.fintypeOfDomOfCoker f = Group.fintypeOfKerLeRange f (QuotientGroup.mk' f.range) ⋯

@[instance_reducible]

noncomputable def AddGroup.fintypeOfDomOfCoker {F : Type u_1} {G : Type u_2} [AddGroup F] [AddGroup G] [Fintype F] (f : F →+ G) [f.range.Normal] [Fintype (G ⧸ f.range)] : Fintype G

If F and coker(F →+ G) are finite, then G is finite.

Equations

• AddGroup.fintypeOfDomOfCoker f = AddGroup.fintypeOfKerLeRange f (QuotientAddGroup.mk' f.range) ⋯

theorem finite_iff_subgroup_quotient {G : Type u_2} [Group G] (H : Subgroup G) : Finite G ↔ Finite ↥H ∧ Finite (G ⧸ H)

theorem finite_iff_addSubgroup_quotient {G : Type u_2} [AddGroup G] (H : AddSubgroup G) : Finite G ↔ Finite ↥H ∧ Finite (G ⧸ H)

theorem Finite.of_subgroup_quotient {G : Type u_2} [Group G] (H : Subgroup G) [Finite ↥H] [Finite (G ⧸ H)] : Finite G

theorem Finite.of_addSubgroup_quotient {G : Type u_2} [AddGroup G] (H : AddSubgroup G) [Finite ↥H] [Finite (G ⧸ H)] : Finite G