Theorems · Definition · group theory

CommGroup.subgroupOrderIsoSubgroupMonoidHom

(G : Type u_1) →
  (M : Type u_2) →
    [inst : CommGroup G] →
      [Finite G] →
        [inst_2 : CommMonoid M] →
          [hM : HasEnoughRootsOfUnity M (Monoid.exponent G)] → Subgroup G ≃o (Subgroup (G →* Mˣ))ᵒᵈ

The order reversing bijection that sends a subgroup of G to its dual subgroup in G →* Mˣ where G is a finite commutative group and M is a commutative monoid with enough nth roots of unity, where n is the exponent of G.

Defined in

Mathlib.GroupTheory.FiniteAbelian.Duality

Cited by

7 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommGroupFiniteCommMonoidHasEnoughRootsOfUnity