Theorems · Theorem · group theory
orderOf_pos_iff
∀ {G : Type u_1} [inst : Monoid G] {x : G}, 0 < orderOf x ↔ IsOfFinOrder xA group element has finite order iff its order is positive.
- Defined in
- Mathlib.GroupTheory.OrderOfElement
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Monoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Monoidstatement and proof · cited by 3,887
- orderOfstatement and proof · cited by 324
- pos_iff_ne_zeroproof · cited by 180
- IsOfFinOrderstatement · cited by 113
- iff_not_commproof · cited by 16
- orderOf_eq_zero_iffproof · cited by 8
Cited by8
Results whose statement or proof uses this declaration.
- Function.Injective.isOfFinOrder_iffproof · cited by 2
- IsOfFinOrder.monoproof · cited by 2
- FiniteField.bijective_frobeniusAlgHom_powproof · cited by 2
- IsPrimitiveRoot.exists_posproof · cited by 2
- NumberField.Units.torsion_eq_one_or_neg_one_of_odd_finrankproof · cited by 1
- Commute.isOfFinOrder_mulproof · cited by 1
- minpoly_algHom_toLinearMapproof · cited by 0
- approxOrderOf.smul_eq_of_mul_dvdproof · cited by 0