Deducing finiteness of a group.

noncomputable def Group.fintypeOfKerLeRange {F G H : Type u} [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.

noncomputable def AddGroup.fintypeOfKerLeRange {F G H : Type u} [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.

noncomputable def Group.fintypeOfKerEqRange {F G H : Type u} [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.

noncomputable def AddGroup.fintypeOfKerEqRange {F G H : Type u} [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.

noncomputable def Group.fintypeOfKerOfCodom {G H : Type u} [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.

noncomputable def AddGroup.fintypeOfKerOfCodom {G H : Type u} [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.

noncomputable def Group.fintypeOfDomOfCoker {F G : Type u} [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.

noncomputable def AddGroup.fintypeOfDomOfCoker {F G : Type u} [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.

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

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