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Mathlib.GroupTheory.QuotientGroup.Finite

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) :

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

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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) :

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

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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) :

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

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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) :

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

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

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

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

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

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

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

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

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

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