Theorems · Theorem · number theory

MulChar.card_subgroupOrderIsoSubgroupMulChar

∀ {M : Type u_1} {R : Type u_2} [inst : CommMonoid M] [inst_1 : CommRing R] [inst_2 : Finite M]
  [inst_3 : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)] {H : Subgroup Mˣ},
  Nat.card ↥(OrderDual.ofDual ((MulChar.subgroupOrderIsoSubgroupMulChar M R) H)) = Nat.card (Mˣ ⧸ H)

The cardinality of the dual subgroup of MulChar M R associated to a subgroup H of Mˣ equals the index of H in Mˣ.

Defined in

Mathlib.NumberTheory.MulChar.Duality

Cited by

1 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoidCommRingFiniteHasEnoughRootsOfUnity