Theorems · Theorem · group theory
Group.isNilpotent_of_product_of_sylow_group
∀ {G : Type u_1} [hG : Group G] [Finite G] (e : ((p : ↥(Nat.card G).primeFactors) → (P : Sylow (↑p) G) → ↥↑P) ≃* G),
Group.IsNilpotent GIf a finite group is the direct product of its Sylow groups, it is nilpotent
- Defined in
- Mathlib.GroupTheory.Nilpotent
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 116 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement and proof · cited by 13,712
- Groupstatement and proof · cited by 6,238
- Subgroupstatement · cited by 3,593
- Finitestatement and proof · cited by 3,029
- Factproof · cited by 2,726
- Nat.Primeproof · cited by 2,059
- MulEquivstatement and proof · cited by 1,142
- Nat.cardstatement and proof · cited by 844
- Nat.primeFactorsstatement and proof · cited by 129
- Sylowstatement and proof · cited by 103
- Sylow.toSubgroupstatement and proof · cited by 86
- Group.IsNilpotentstatement and proof · cited by 80
Cited by2
Results whose statement or proof uses this declaration.
- Group.isNilpotent_of_finite_tfaeproof · cited by 2
- isNilpotent_of_product_of_sylow_groupproof · cited by 0