Theorems · Theorem · group theory
orderOf_injective
∀ {G : Type u_1} [inst : Monoid G] {H : Type u_6} [inst_1 : Monoid H] (f : G →* H),
Function.Injective ⇑f → ∀ (x : G), orderOf (f x) = orderOf xAn injective homomorphism of monoids preserves orders of elements.
- Defined in
- Mathlib.GroupTheory.OrderOfElement
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- DFunLike.coestatement and proof · cited by 62,936
- Monoidstatement and proof · cited by 3,887
- MonoidHomstatement and proof · cited by 3,629
- orderOfstatement · cited by 324
- MonoidHom.map_powproof · cited by 30
- MonoidHom.map_oneproof · cited by 27
Cited by12
Results whose statement or proof uses this declaration.
- orderOf_unitsproof · cited by 9
- Subgroup.orderOf_coeproof · cited by 5
- orderOf_submonoidproof · cited by 5
- Function.Injective.isOfFinOrder_iffproof · cited by 2
- lucas_primalityproof · cited by 2
- ZMod.isCyclic_units_of_prime_powproof · cited by 2
- MulEquiv.orderOf_eqproof · cited by 1
- reverse_lucas_primalityproof · cited by 1
- IsCyclotomicExtension.Rat.inertiaDeg_eq_of_not_dvdproof · cited by 1
- orderOf_piMulSingleproof · cited by 0
- IsCyclotomicExtension.Rat.galEquivZMod_stabilizerproof · cited by 0
- minpoly_algHom_toLinearMapproof · cited by 0