Theorems · Theorem · group theory

CommGroup.monoidHom_mulEquiv_of_hasEnoughRootsOfUnity

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

A finite commutative group G is (noncanonically) isomorphic to the group G →* Mˣ when 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

2 results in Mathlib

Foundations

Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommGroupFiniteCommMonoidHasEnoughRootsOfUnity