Theorems · Theorem · group theory
Sylow.commutator_eq_bot_or_commutator_eq_self
∀ {G : Type u_1} [inst : Group G] {p : ℕ} [Fact (Nat.Prime p)] [Finite G] (P : Sylow p G) [IsCyclic ↥↑P] [(↑P).Normal]
{K : Subgroup G}, K.IsComplement' ↑P → ⁅K, ↑P⁆ = ⊥ ∨ ⁅K, ↑P⁆ = ↑PIf a normal cyclic Sylow p-subgroup P has a complement K, then either ⁅K, P⁆ = ⊥ or
⁅K, P⁆ = P.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Groupstatement and proof · cited by 6,238
- Bot.botstatement · cited by 4,720
- Subgroupstatement and proof · cited by 3,593
- Finitestatement and proof · cited by 3,029
- Factstatement and proof · cited by 2,726
- Nat.Primestatement and proof · cited by 2,059
- Bracket.bracketstatement · cited by 642
- le_topproof · cited by 411
- Subgroup.Normalstatement and proof · cited by 334
- IsCyclicstatement and proof · cited by 122
- Sylowstatement and proof · cited by 103
- Sylow.toSubgroupstatement and proof · cited by 86
Cited by1
Results whose statement or proof uses this declaration.
- Sylow.le_center_or_le_commutatorproof · cited by 1