Theorems · Definition · group theory
Subgroup.IsComplement
{G : Type u_1} → [Group G] → Set G → Set G → PropS and T are complements if (*) : S × T → G is a bijection.
This notion generalizes left transversals, right transversals, and complementary subgroups.
If S and T are SetLikes such as Subgroups, see isComplement_iff_bijective for a
more ergonomic way to unfold.
- Defined in
- Mathlib.GroupTheory.Complement
- Cited by
- 94 results in Mathlib
- Foundations
- Depth 6 from the axioms, rests on 45 definitions · uses no axioms
- Assumes
- Group
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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
- Set.Elemproof · cited by 7,166
- Groupstatement and proof · cited by 6,238
- Function.Bijectiveproof · cited by 863
Cited by112
Results whose statement or proof uses this declaration.
- Subgroup.IsComplement.equivstatement and proof · cited by 34
- Subgroup.IsComplement'proof · cited by 34
- Subgroup.IsComplement.leftQuotientEquivstatement and proof · cited by 17
- Subgroup.LeftTransversalproof · cited by 13
- Subgroup.IsComplement.equiv_fst_eq_mul_invstatement and proof · cited by 8
- Subgroup.IsComplement.toRightFunstatement and proof · cited by 8
- Subgroup.IsComplement.equiv_snd_eq_inv_mulstatement and proof · cited by 8
- Monoid.PushoutI.NormalWord.Transversal.complstatement · cited by 7
- Subgroup.IsComplement.mul_inv_toRightFun_memstatement and proof · cited by 6
- Subgroup.IsComplement.equiv_fst_mul_equiv_sndstatement and proof · cited by 5
- Subgroup.IsComplement.mul_eqstatement and proof · cited by 5
- Subgroup.IsComplement.rightQuotientEquivstatement and proof · cited by 5