Theorems · Definition · group theory

CommGroup.monoidHomMonoidHomEquiv

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

The MulEquiv between the double dual (G →* Mˣ) →* Mˣ of a finite commutative group G and itself where M is a commutative monoid with enough nth roots of unity, where n is the exponent of G. The image g of η : (G →* Mˣ) →* Mˣ is such that, for all φ : G →* Mˣ, we have φ g = η g, see CommGroup.apply_monoidHomMonoidHomEquiv.

Defined in

Mathlib.GroupTheory.FiniteAbelian.Duality

Cited by

5 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommGroupFiniteCommMonoidHasEnoughRootsOfUnity