Theorems · Definition · group theory
Subgroup.IsComplement.leftQuotientEquiv
{G : Type u_1} → [inst : Group G] → {H : Subgroup G} → {S : Set G} → Subgroup.IsComplement S ↑H → G ⧸ H ≃ ↑SA left transversal is in bijection with left cosets.
- Defined in
- Mathlib.GroupTheory.Complement
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Group
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.
- Setstatement and proof · cited by 53,352
- Equivstatement · cited by 8,337
- SetLike.coestatement and proof · cited by 8,199
- Set.Elemstatement · cited by 7,166
- Groupstatement and proof · cited by 6,238
- Equiv.symmproof · cited by 3,681
- Subgroupstatement and proof · cited by 3,593
- HasQuotient.Quotientstatement · cited by 2,301
- Set.domRestrictproof · cited by 383
- QuotientGroup.mkproof · cited by 196
- Subgroup.IsComplementstatement and proof · cited by 94
- Equiv.ofBijectiveproof · cited by 70
Cited by20
Results whose statement or proof uses this declaration.
- Subgroup.leftTransversals.diffproof · cited by 8
- Subgroup.IsComplement.toLeftFunproof · cited by 4
- Subgroup.smul_apply_eq_smul_apply_inv_smulstatement and proof · cited by 4
- Subgroup.IsComplement'.QuotientMulEquivproof · cited by 3
- Subgroup.leftTransversals.diff_mul_diffproof · cited by 3
- Subgroup.IsComplement.finite_left_iffproof · cited by 3
- Subgroup.leftCoset_cover_filter_FiniteIndex_auxproof · cited by 3
- Subgroup.smul_diff'proof · cited by 2
- Subgroup.transferTransversal_applystatement · cited by 2
- MonoidHom.transfer_eq_prod_quotient_orbitRel_zpowers_quotproof · cited by 2
- Subgroup.smul_leftQuotientEquivstatement · cited by 1
- Subgroup.leftTransversals.smul_diff_smulproof · cited by 1