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Theorems · Theorem · group theory

CommGroup.card_domRestrictHom_ker

∀ {G : Type u_1} (M : Type u_2) [inst : CommGroup G] [Finite G] [inst_2 : CommMonoid M]
  [hM : HasEnoughRootsOfUnity M (Monoid.exponent G)] (H : Subgroup G),
  Nat.card ↥(MonoidHom.domRestrictHom H Mˣ).ker = Nat.card (G ⧸ H)
Defined in
Mathlib.GroupTheory.FiniteAbelian.Duality
Cited by
2 results in Mathlib
Foundations
Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CommGroupFiniteCommMonoidHasEnoughRootsOfUnity
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

CommGroup.card_subgroupOrderIsoSubgroupMonoidHom · cited by 1CommGroup.card_subgroupOr…CommGroup.card_restrictHom_ker · cited by 0CommGroup.card_restrictHo…MonoidHom · cited by 3629MonoidHomSubgroup · cited by 3593SubgroupFinite · cited by 3029FiniteUnits · cited by 2804UnitsHasQuotient.Quotient · cited by 2301HasQuotient.QuotientCommMonoid · cited by 2264CommMonoidCommGroup · cited by 990CommGroupNat.card · cited by 844Nat.cardMonoidHom.ker · cited by 212MonoidHom.kerNat.card_congr · cited by 133Nat.card_congrMonoid.exponent · cited by 128Monoid.exponentMulEquiv.toEquiv · cited by 126MulEquiv.toEquivHasEnoughRootsOfUnity · cited by 56HasEnoughRootsOfUnityMonoidHom.domRestrictHom · cited by 12MonoidHom.domRestrictHomMonoidHom.domRestrictHomKerEquiv · cited by 6MonoidHom.domRestrictHomK…CommGroup.card_domRestrictHom…CITED BYCITES

Cites18

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by2

Results whose statement or proof uses this declaration.