Theorems · Theorem · group theory

CommGroup.card_subgroupOrderIsoSubgroupMonoidHom

∀ {G : Type u_1} (M : Type u_2) [inst : CommGroup G] [inst_1 : Finite G] [inst_2 : CommMonoid M]
  [hM : HasEnoughRootsOfUnity M (Monoid.exponent G)] (H : Subgroup G),
  Nat.card ↥(OrderDual.ofDual ((CommGroup.subgroupOrderIsoSubgroupMonoidHom G M) H)) = Nat.card (G ⧸ H)

The cardinality of the dual subgroup of G →* Mˣ associated to a subgroup H of G equals the index of H in G.

Defined in

Mathlib.GroupTheory.FiniteAbelian.Duality

Cited by

1 results in Mathlib

Foundations

Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommGroupFiniteCommMonoidHasEnoughRootsOfUnity