Theorems · Theorem · group theory
Subgroup.nilpotent_iff_lowerCentralSeries
∀ {G : Type u_1} [inst : Group G], Group.IsNilpotent G ↔ ∃ n, ⊤.lowerCentralSeries n = ⊥A group is nilpotent if and only if its lower central series eventually reaches the trivial subgroup.
- Defined in
- Mathlib.GroupTheory.Nilpotent
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 95 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.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Top.topstatement and proof · cited by 9,680
- Groupstatement and proof · cited by 6,238
- Bot.botstatement · cited by 4,720
- Subgroupstatement · cited by 3,593
- Group.IsNilpotentstatement · cited by 80
- Subgroup.lowerCentralSeriesstatement · cited by 57
- Subgroup.isNilpotent_iff_lowerCentralSeriesproof · cited by 2
- Group.isNilpotent_topproof · cited by 1
Cited by9
Results whose statement or proof uses this declaration.
- Subgroup.lowerCentralSeries_length_eq_nilpotencyClassstatement and proof · cited by 5
- Subgroup.lowerCentralSeries_nilpotencyClassproof · cited by 3
- Group.nilpotencyClass_le_of_ker_le_centerproof · cited by 2
- Subgroup.isNilpotent_of_ker_le_centerproof · cited by 2
- Group.isNilpotent_pi_of_bounded_classproof · cited by 1
- Subgroup.lowerCentralSeries_eq_bot_iff_upperCentralSeries_eq_topproof · cited by 1
- Subgroup.nilpotencyClass_leproof · cited by 0
- nilpotent_iff_lowerCentralSeriesproof · cited by 0
- lowerCentralSeries_length_eq_nilpotencyClassstatement · cited by 0