Weak chain conditions on subgroups

DefinitionWeak maximal condition on subgroups#✓fully proved

A group \(G\) satisfies the weak maximal condition on subgroups when there is no infinite ascending chain \(H_0 \le H_1 \le \cdots\) of subgroups of \(G\) in which every index \(|H_{i+1} : H_i|\) is infinite.

def WeakMaximalCondition (G : Type*) [Group G] : Prop :=
  ¬ ∃ H : ℕ → Subgroup G, ∀ i, H i ≤ H (i + 1) ∧ (H i).relIndex (H (i + 1)) = 0

Used by 1

Used byMinimal condition implies weak maximal condition

DefinitionWeak minimal condition on subgroups#✓fully proved

The weak minimal condition on subgroups: there is no infinite descending chain H₀ ≥ H₁ ≥ ⋯ of subgroups with every index |Hᵢ : Hᵢ₊₁| infinite.

def WeakMinimalCondition (G : Type*) [Group G] : Prop :=
  ¬ ∃ H : ℕ → Subgroup G, ∀ i, H (i + 1) ≤ H i ∧ (H (i + 1)).relIndex (H i) = 0

Used by 1

Used byMaximal condition implies weak minimal condition

DefinitionMinimal condition implies weak maximal condition#✓fully proved

Kourovka 10.14(a) (D. I. Zaitsev): every group with the minimal condition on subgroups satisfies the weak maximal condition on subgroups.

def minimalImpliesWeakMaximal : Prop :=
  ∀ (G : Type) [Group G], WellFoundedLT (Subgroup G) → WeakMaximalCondition G

Uses 1 · Used by 1

Statement usesWeak maximal condition on subgroups

Used byA group with the minimal condition can fail the weak maximal condition

DefinitionMaximal condition implies weak minimal condition#✓fully proved

Kourovka 10.14(b): every group with the maximal condition on subgroups satisfies the weak minimal condition on subgroups.

def maximalImpliesWeakMinimal : Prop :=
  ∀ (G : Type) [Group G], WellFoundedGT (Subgroup G) → WeakMinimalCondition G

Uses 1 · Used by 1

Statement usesWeak minimal condition on subgroups

Used byA group with the maximal condition can fail the weak minimal condition

TheoremA group with the minimal condition can fail the weak maximal condition#●ready to prove

There is a group \(G\) satisfying the minimal condition on subgroups (there is no infinite strictly descending chain of subgroups of \(G\)) that has an infinite ascending chain \(H_0 \le H_1 \le \cdots\) of subgroups in which every index \(|H_{i+1} : H_i|\) is infinite. This answers Kourovka Problem 10.14(a) negatively.

The Kourovka Archive answers 10.14(a) "No, not always", citing Obraztsov (1990).

public theorem_wanted minimalImpliesWeakMaximal_refuted : ¬ minimalImpliesWeakMaximal

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 10.14 (D. I. Zaitsev); resolution: V. N. Obraztsov, Math. USSR-Sb. 66, no. 2 (1990), 541-553 (a); V. N. Obraztsov, Siberian Math. J. 32, no. 1 (1991), 79-84 (b), Kourovka Archive, Problem 10.14 (both parts "No, not always").

Uses 1

Statement usesMinimal condition implies weak maximal condition

TheoremA group with the maximal condition can fail the weak minimal condition#●ready to prove

There is a group \(G\) satisfying the maximal condition on subgroups (there is no infinite strictly ascending chain of subgroups of \(G\)) that has an infinite descending chain \(H_0 \ge H_1 \ge \cdots\) of subgroups in which every index \(|H_i : H_{i+1}|\) is infinite. This answers Kourovka Problem 10.14(b) negatively.

The Kourovka Archive answers 10.14(b) "No, not always", citing Obraztsov (1991).

public theorem_wanted maximalImpliesWeakMinimal_refuted : ¬ maximalImpliesWeakMinimal

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 10.14 (D. I. Zaitsev); resolution: V. N. Obraztsov, Math. USSR-Sb. 66, no. 2 (1990), 541-553 (a); V. N. Obraztsov, Siberian Math. J. 32, no. 1 (1991), 79-84 (b), Kourovka Archive, Problem 10.14 (both parts "No, not always").

Uses 1

Statement usesMaximal condition implies weak minimal condition

4 fully proved2 ready to prove

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