Theorems · Definition · group theory

MulAction.zpowersQuotientStabilizerEquiv

{α : Type u_3} →
  {β : Type u_4} →
    [inst : Group α] →
      (a : α) →
        [inst_1 : MulAction α β] →
          (b : β) →
            ↥(Subgroup.zpowers a) ⧸ MulAction.stabilizer (↥(Subgroup.zpowers a)) b ≃*
              Multiplicative (ZMod (Function.minimalPeriod (fun x => a • x) b))

The quotient (a ^ ℤ) ⧸ (stabilizer b) is cyclic of order minimalPeriod ((•) a) b.

Defined in

Mathlib.Data.ZMod.QuotientGroup

Cited by

1 results in Mathlib

Foundations

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

Assumes

GroupMulAction