Theorems · Theorem · group theory
MonoidHom.ker_eq_bot_iff
∀ {G : Type u_1} [inst : Group G] {M : Type u_7} [inst_1 : MulOneClass M] (f : G →* M),
f.ker = ⊥ ↔ Function.Injective ⇑f- Defined in
- Mathlib.Algebra.Group.Subgroup.Ker
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- GroupMulOneClass
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Groupstatement and proof · cited by 6,238
- Bot.botstatement and proof · cited by 4,720
- MonoidHomstatement and proof · cited by 3,629
- Subgroupstatement and proof · cited by 3,593
- MulOneClassstatement and proof · cited by 1,018
- MonoidHom.kerstatement and proof · cited by 212
- bot_uniqueproof · cited by 57
- MonoidHom.map_oneproof · cited by 27
- inv_mul_eq_oneproof · cited by 11
- Subgroup.mem_botproof · cited by 11
- MonoidHom.eq_iffproof · cited by 2
Cited by16
Results whose statement or proof uses this declaration.
- MonoidHom.ker_eq_botproof · cited by 10
- MonoidHom.ker_transferSylow_isComplement'proof · cited by 3
- zpowersHom_bijectiveproof · cited by 2
- IsQuotientCoveringMap.monodromyPerm_injectiveproof · cited by 2
- MonoidHom.injective_noncommPiCoprod_of_iSupIndepproof · cited by 2
- GrpCat.mono_iff_injectiveproof · cited by 2
- CommGrpCat.mono_iff_ker_eq_botproof · cited by 1
- Group.isCyclic_of_coprime_card_range_card_kerproof · cited by 1
- isZGroup_of_coprimeproof · cited by 1
- CategoryTheory.PreGaloisCategory.toAut_injective_of_non_trivialproof · cited by 1
- GrpCat.mono_iff_ker_eq_botproof · cited by 1
- IsQuotientCoveringMap.fundamentalGroupToMulOpposite_injectiveproof · cited by 1