Theorems · Theorem · group theory
Subgroup.isNilpotent_iff_lowerCentralSeries
∀ {G : Type u_1} [inst : Group G] (S : Subgroup G), Group.IsNilpotent ↥S ↔ ∃ n, S.lowerCentralSeries n = ⊥A subgroup is nilpotent iff its lower central series (computed in the ambient group) eventually vanishes.
- Defined in
- Mathlib.GroupTheory.Nilpotent
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Group
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.
- Top.topproof · cited by 9,680
- Groupstatement and proof · cited by 6,238
- Bot.botstatement and proof · cited by 4,720
- Subgroupstatement and proof · cited by 3,593
- Subgroup.mapproof · cited by 301
- Subgroup.subtypeproof · cited by 185
- le_bot_iffproof · cited by 116
- Group.IsNilpotentstatement · cited by 80
- Subgroup.lowerCentralSeriesstatement and proof · cited by 57
- Subgroup.IsDescendingCentralSeriesproof · cited by 14
- Subgroup.map_subtype_injproof · cited by 11
- Subgroup.map_botproof · cited by 10
Cited by2
Results whose statement or proof uses this declaration.
- Subgroup.nilpotent_iff_lowerCentralSeriesproof · cited by 9
- Subgroup.isNilpotent_of_lowerCentralSeries_eq_botproof · cited by 0