Theorems · Theorem · group theory
AddSubgroup.IsComplement.finite_left_iff
∀ {G : Type u_1} [inst : AddGroup G] {H : AddSubgroup G} {S : Set G},
AddSubgroup.IsComplement S ↑H → (Finite ↑S ↔ H.FiniteIndex)A left transversal is finite iff the subgroup has finite index.
- Defined in
- Mathlib.GroupTheory.Complement
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddGroup
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
- SetLike.coestatement and proof · cited by 8,199
- Set.Elemstatement · cited by 7,166
- AddGroupstatement and proof · cited by 4,410
- AddSubgroupstatement and proof · cited by 3,232
- Finitestatement and proof · cited by 3,029
- HasQuotient.Quotientproof · cited by 2,301
- AddSubgroup.FiniteIndexstatement and proof · cited by 66
- AddSubgroup.IsComplementstatement and proof · cited by 57
- Equiv.finite_iffproof · cited by 21
- AddSubgroup.IsComplement.leftQuotientEquivproof · cited by 8
- AddSubgroup.finiteIndex_of_finite_quotientproof · cited by 7
Cited by3
Results whose statement or proof uses this declaration.
- AddSubgroup.exists_leftTransversal_of_FiniteIndexproof · cited by 1
- AddSubgroup.IsComplement.finite_leftproof · cited by 1
- MeasureTheory.AddSubgroup.index_mul_measureproof · cited by 0