Theorems · Theorem · group theory
Subgroup.nilpotent_iff_finite_descending_central_series
∀ (G : Type u_1) [inst : Group G], Group.IsNilpotent G ↔ ∃ n H, Subgroup.IsDescendingCentralSeries H ∧ H n = ⊥
A group G is nilpotent iff there exists a descending central series which reaches the
trivial group in a finite time.
- Defined in
- Mathlib.GroupTheory.Nilpotent
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 92 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.
Cites11
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
- tsub_selfproof · cited by 154
- Group.IsNilpotentstatement · cited by 80
- Subgroup.IsAscendingCentralSeriesproof · cited by 14
- Subgroup.IsDescendingCentralSeriesstatement and proof · cited by 14
- Subgroup.nilpotent_iff_finite_ascending_central_seriesproof · cited by 4
- Subgroup.is_ascending_rev_series_of_is_descendingproof · cited by 3
- Subgroup.is_descending_rev_series_of_is_ascendingproof · cited by 3
Cited by4
Results whose statement or proof uses this declaration.
- Subgroup.isNilpotent_iff_lowerCentralSeriesproof · cited by 2
- Subgroup.least_descending_central_series_length_eq_nilpotencyClassstatement and proof · cited by 2
- least_descending_central_series_length_eq_nilpotencyClassstatement · cited by 0
- nilpotent_iff_finite_descending_central_seriesproof · cited by 0