Theorems · Definition · group theory

AddAction.orbitZMultiplesEquiv

{α : Type u_5} →
  {β : Type u_6} →
    [inst : AddGroup α] →
      (a : α) →
        [inst_1 : AddAction α β] →
          (b : β) →
            ↑(AddAction.orbit (↥(AddSubgroup.zmultiples a)) b) ≃ ZMod (Function.minimalPeriod (fun x => a +ᵥ x) b)

The orbit (ℤ • a) +ᵥ b is a cycle of order minimalPeriod (a +ᵥ ·) b.

Defined in

Mathlib.Data.ZMod.QuotientGroup

Cited by

4 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddGroupAddAction