Theorems · Definition · group theory

AddAction.zmultiplesQuotientStabilizerEquiv

{α : Type u_3} →
  {β : Type u_4} →
    [inst : AddGroup α] →
      (a : α) →
        [inst_1 : AddAction α β] →
          (b : β) →
            ↥(AddSubgroup.zmultiples a) ⧸ AddAction.stabilizer (↥(AddSubgroup.zmultiples a)) b ≃+
              ZMod (Function.minimalPeriod (fun x => a +ᵥ x) b)

The quotient (ℤ ∙ a) ⧸ (stabilizer b) is cyclic of order minimalPeriod (a +ᵥ ·) b.

Defined in

Mathlib.Data.ZMod.QuotientGroup

Cited by

1 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddGroupAddAction