Theorems · Theorem · group theory
isNilpotent_of_product_of_sylow_group
Deprecated since 2026-03-25Use Group.isNilpotent_of_product_of_sylow_group instead.
∀ {G : Type u_1} [hG : Group G] [Finite G] (e : ((p : ↥(Nat.card G).primeFactors) → (P : Sylow (↑p) G) → ↥↑P) ≃* G),
Group.IsNilpotent GAlias of Group.isNilpotent_of_product_of_sylow_group.
If a finite group is the direct product of its Sylow groups, it is nilpotent
- Defined in
- Mathlib.GroupTheory.Nilpotent
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement · cited by 13,712
- Groupstatement · cited by 6,238
- Subgroupstatement · cited by 3,593
- Finitestatement · cited by 3,029
- MulEquivstatement · cited by 1,142
- Nat.cardstatement · cited by 844
- Nat.primeFactorsstatement · cited by 129
- Sylowstatement · cited by 103
- Sylow.toSubgroupstatement · cited by 86
- Group.IsNilpotentstatement · cited by 80
- Group.isNilpotent_of_product_of_sylow_groupproof · cited by 2
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