Theorems · Definition · group theory
unitsCenterToCenterUnits
(M : Type u_1) → [inst : Monoid M] → (↥(Submonoid.center M))ˣ →* ↥(Submonoid.center Mˣ)
For a monoid, the units of the center inject into the center of the units. This is not an
equivalence in general; one case where this holds is for groups with zero, which is covered in
centerUnitsEquivUnitsCenter.
- Defined in
- Mathlib.GroupTheory.Submonoid.Center
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 28 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.
Cites8
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
- MonoidHomstatement · cited by 3,629
- Submonoidstatement · cited by 3,086
- Unitsstatement and proof · cited by 2,804
- Units.mapproof · cited by 95
- Submonoid.subtypeproof · cited by 26
- Submonoid.centerstatement and proof · cited by 24
- MonoidHom.codRestrictproof · cited by 8
Cited by3
Results whose statement or proof uses this declaration.
- Subgroup.centerUnitsEquivUnitsCenterproof · cited by 2
- val_unitsCenterToCenterUnits_apply_coestatement and proof · cited by 0
- unitsCenterToCenterUnits_injectivestatement and proof · cited by 0