Theorems · Theorem · group theory
MonoidHom.map_closure
∀ {G : Type u_1} [inst : Group G] {N : Type u_5} [inst_1 : Group N] (f : G →* N) (s : Set G),
Subgroup.map f (Subgroup.closure s) = Subgroup.closure (⇑f '' s)The image under a monoid homomorphism of the subgroup generated by a set equals the subgroup generated by the image of the set.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Map
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement and proof · cited by 53,352
- Groupstatement and proof · cited by 6,238
- Set.imagestatement · cited by 5,609
- MonoidHomstatement and proof · cited by 3,629
- Subgroupstatement and proof · cited by 3,593
- Subgroup.mapstatement · cited by 301
- Subgroup.closurestatement · cited by 196
- GaloisInsertion.gcproof · cited by 137
- Set.image_preimageproof · cited by 16
- GaloisConnection.l_comm_of_u_commproof · cited by 15
- Subgroup.gc_map_comapproof · cited by 12
Cited by16
Results whose statement or proof uses this declaration.
- MonoidHom.map_zpowersproof · cited by 5
- Group.rank_le_of_surjectiveproof · cited by 3
- surjective_of_isSwap_of_isPretransitive'proof · cited by 2
- NumberField.Units.isMaxRank_iff_closure_finiteIndexproof · cited by 1
- CoxeterSystem.subgroup_closure_range_simpleproof · cited by 1
- PresentedGroup.closure_range_ofproof · cited by 1
- Subgroup.le_normalizer_closure_iffproof · cited by 1
- Subgroup.focalSubgroupOf_eq_closureproof · cited by 1
- Subgroup.closure_mul_image_eq_topproof · cited by 1
- NumberField.IsCMField.closure_realFundSystem_sup_torsionproof · cited by 1
- card_commutator_closureCommutatorRepresentativesproof · cited by 1
- Subgroup.smul_closureproof · cited by 0