Schmidt subgroups

DefinitionMinimal non-nilpotent subgroup#✓fully proved

A subgroup \(M\) of a group \(G\) is minimal non-nilpotent if \(M\) is not nilpotent but every proper subgroup of \(M\) is nilpotent.

def IsMinimalNonNilpotent {G : Type*} [Group G] (M : Subgroup G) : Prop :=
  ¬ Group.IsNilpotent M ∧ ∀ K : Subgroup G, K < M → Group.IsNilpotent K

Used by 1

Used bySubgroups permuting with all minimal non-nilpotent subgroups (Kourovka 17.56)

DefinitionSubgroups permuting with all minimal non-nilpotent subgroups (Kourovka 17.56)#✓fully proved

The statement that if \(H\) is a subgroup of a finite group \(G\) with \(HM = MH\) (as sets of products) for every minimal non-nilpotent subgroup \(M\) of \(G\), then the quotient \(H/H_G\) is nilpotent, where the normal core \(H_G\) is the largest normal subgroup of \(G\) contained in \(H\).

The question is due to V. N. Knyagina and V. S. Monakhov.

def permutesMinimalNonNilpotentCoreQuotientNilpotent : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (H : Subgroup G),
    (∀ M : Subgroup G, IsMinimalNonNilpotent M → (H : Set G) * M = M * H) →
      Group.IsNilpotent (H ⧸ H.normalCore.subgroupOf H)

Uses 1 · Used by 1

Statement usesMinimal non-nilpotent subgroup

Used byPermutability with minimal non-nilpotent subgroups does not force a nilpotent core quotient

TheoremPermutability with minimal non-nilpotent subgroups does not force a nilpotent core quotient#●ready to prove

There exist a finite group \(G\) and a subgroup \(H\) of \(G\) with \(HM = MH\) (as sets of products) for every minimal non-nilpotent subgroup \(M\) of \(G\) such that the quotient \(H/H_G\) is not nilpotent, where the normal core \(H_G\) is the largest normal subgroup of \(G\) contained in \(H\).

This answers Kourovka Problem 17.56 in the negative (Ballester-Bolinches, Esteban-Romero, Kamornikov and Pérez-Calabuig, 2022).

public theorem_wanted permutesMinimalNonNilpotentCoreQuotientNilpotent_refuted :
    ¬ permutesMinimalNonNilpotentCoreQuotientNilpotent

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 17.56 (V. N. Knyagina, V. S. Monakhov); resolution: A. Ballester-Bolinches, R. Esteban-Romero, S. F. Kamornikov and V. Pérez-Calabuig, J. Algebra 608 (2022), 382-387, the Kourovka Notebook (21st issue) marks 17.56 solved (No, not always).

Uses 1

Statement usesSubgroups permuting with all minimal non-nilpotent subgroups (Kourovka 17.56)

DefinitionSchmidt subgroup#✓fully proved

A subgroup \(H\) of a group \(G\) is a Schmidt subgroup when \(H\) is not nilpotent but every proper subgroup of \(H\) is nilpotent.

Finiteness of \(H\) is not part of this definition.

def IsSchmidtSubgroup {G : Type} [Group G] (H : Subgroup G) : Prop :=
  ¬ Group.IsNilpotent H ∧ ∀ K : Subgroup G, K < H → Group.IsNilpotent K

Used by 2

Used byClaim of Kourovka Problem 19.85 · Claim of Kourovka Problem 18.30

Definition\(\mathbb{P}\)-subnormal subgroup#✓fully proved

A subgroup \(H\) of a group \(G\) is \(\mathbb{P}\)-subnormal in \(G\) when \(H = G\) or there is a chain of subgroups \(H = H_0 < H_1 < \cdots < H_n = G\) in which every index \(|H_i : H_{i-1}|\) is a prime number.

def IsPSubnormal {G : Type} [Group G] (H : Subgroup G) : Prop :=
  Relation.ReflTransGen (fun A B : Subgroup G => A ≤ B ∧ (A.relIndex B).Prime) H ⊤

Used by 1

Used byClaim of Kourovka Problem 18.30

DefinitionClaim of Kourovka Problem 18.30#✓fully proved

The statement that every finite group in which every Schmidt subgroup is \(\mathbb{P}\)-subnormal is soluble. The problem is due to A. F. Vasil'ev, T. I. Vasil'eva and V. N. Tyutyanov.

def schmidtSubgroupsPSubnormalImpliesSoluble : Prop :=
  ∀ (G : Type) [Group G] [Finite G],
    (∀ H : Subgroup G, IsSchmidtSubgroup H → IsPSubnormal H) → Group.IsSolvable G

Uses 2 · Used by 1

Statement uses\(\mathbb{P}\)-subnormal subgroup · Schmidt subgroup

Used byFinite groups with \(\mathbb{P}\)-subnormal Schmidt subgroups are soluble

TheoremFinite groups with \(\mathbb{P}\)-subnormal Schmidt subgroups are soluble#●ready to prove

Every finite group in which every Schmidt subgroup is \(\mathbb{P}\)-subnormal is soluble. This answers Kourovka Notebook Problem 18.30 in the affirmative.

public theorem_wanted schmidtSubgroupsPSubnormalImpliesSoluble_holds :
    schmidtSubgroupsPSubnormalImpliesSoluble

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 18.30 (A. F. Vasil'ev, T. I. Vasil'eva, V. N. Tyutyanov); resolution: V. N. Tyutyanov, Probl. Fiz. Mat. Tekhn. 2015, no. 1 (22), 88-91 (Russian), Kourovka Archive, Problem 18.30: Yes, it must, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesClaim of Kourovka Problem 18.30

DefinitionClaim of Kourovka Problem 19.85#✓fully proved

The statement that for every partition \(\sigma\) of the set of all primes into nonempty pairwise disjoint blocks and every finite group \(G\) in which every Schmidt subgroup is \(\sigma\)-subnormal, \(G\) has a normal \(\sigma\)-nilpotent subgroup \(N\) such that \(G / N\) is cyclic: there is \(g \in G\) such that every \(x \in G\) satisfies \(x g^{-k} \in N\) for some integer \(k\). The problem is due to A. N. Skiba.

def schmidtSigmaSubnormalCyclicQuotient : Prop :=
  ∀ (I : Type) (σ : I → Set ℕ), IsPrimePartition σ → ∀ (G : Type) [Group G] [Finite G],
    (∀ S : Subgroup G, IsSchmidtSubgroup S → IsSigmaSubnormal σ S) →
    ∃ N : Subgroup G, N.Normal ∧ IsSigmaNilpotent σ N ∧
      ∃ g : G, ∀ x : G, ∃ k : ℤ, x * (g ^ k)⁻¹ ∈ N

Uses 4 · Used by 1

Statement usesPartitions of the primes · \(\sigma\)-nilpotent groups · \(\sigma\)-subnormal subgroups · Schmidt subgroup

Used byFinite groups with \(\sigma\)-subnormal Schmidt subgroups are \(\sigma\)-nilpotent-by-cyclic

TheoremFinite groups with \(\sigma\)-subnormal Schmidt subgroups are \(\sigma\)-nilpotent-by-cyclic#●ready to prove

Let \(\sigma\) be a partition of the set of all primes into nonempty pairwise disjoint blocks, and let \(G\) be a finite group in which every Schmidt subgroup is \(\sigma\)-subnormal. Then \(G\) has a normal \(\sigma\)-nilpotent subgroup \(N\) such that \(G / N\) is cyclic: there is \(g \in G\) such that every \(x \in G\) satisfies \(x g^{-k} \in N\) for some integer \(k\). This answers Kourovka Notebook Problem 19.85 in the affirmative.

Yi and Kamornikov prove that \(G / F_\sigma(G)\) is cyclic, where \(F_\sigma(G)\) is the largest normal \(\sigma\)-nilpotent subgroup of \(G\); for nilpotent \(G\) one takes \(N = G\). The case of the partition of the primes into single primes was known before.

public theorem_wanted schmidtSigmaSubnormalCyclicQuotient_holds :
    schmidtSigmaSubnormalCyclicQuotient

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 19.85 (A. N. Skiba); resolution: X. Yi and S. F. Kamornikov, Finite groups with σ-subnormal Schmidt subgroups, J. Algebra 560 (2020), 181-191, main result, quoted as Lemma 8 (reference [11]) in A. Ballester-Bolinches, S. F. Kamornikov and X. Yi, Finite groups with σ-subnormal Schmidt subgroups, Bull. Malays. Math. Sci. Soc. (2022): if every Schmidt subgroup of a non-nilpotent group G is σ-subnormal, then G / F_σ(G) is cyclic, F_σ(G) the largest normal σ-nilpotent subgroup; Kourovka Archive, Problem 19.85: Yes, it is true, https://doi.org/10.1007/s40840-022-01369-y.

Uses 1

Statement usesClaim of Kourovka Problem 19.85

6 fully proved3 ready to prove

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