Derived and Frattini subgroups of p-groups

DefinitionDerived subgroups of p-groups as Frattini subgroups (Kourovka 16.11)#✓fully proved

The conjecture of Ya. G. Berkovich that for every prime \(p\) and every finite \(p\)-group \(G\) there is a finite \(p\)-group \(H\) whose Frattini subgroup \(\Phi(H)\), the intersection of the maximal subgroups of \(H\), is isomorphic to the derived subgroup \([G, G]\) of \(G\).

def derivedSubgroupIsFrattini : Prop :=
  ∀ p : ℕ, p.Prime → ∀ (G : Type) [Group G] [Finite G], IsPGroup p G →
    ∃ (H : Type) (_ : Group H) (_ : Finite H), IsPGroup p H ∧
      Nonempty (frattini H ≃* commutator G)

Used by 1

Used byDerived subgroups of p-groups need not be Frattini subgroups of p-groups

TheoremDerived subgroups of p-groups need not be Frattini subgroups of p-groups#●ready to prove

There exist a prime \(p\) and a finite \(p\)-group \(G\) such that for every finite \(p\)-group \(H\), the Frattini subgroup \(\Phi(H)\) of \(H\) is not isomorphic to the derived subgroup \([G, G]\) of \(G\). That is, the conjecture of Kourovka Problem 16.11 is false.

Ionin and Semidetnov take \(p = 2\) and \(G\) the group SmallGroup(256, 511), whose derived subgroup is the Hall-Senior group 32/40 (a GAP computation); by a theorem of van der Waall and de Nijs, that group is not isomorphic to the Frattini subgroup of any finite 2-group. The authors state that their solutions were found with the assistance of large language models and checked by them.

public theorem_wanted derivedSubgroupIsFrattini_refuted : ¬ derivedSubgroupIsFrattini

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 16.11 (Ya. G. Berkovich); resolution: V. Ionin and A. Semidetnov, On some more problems from the Kourovka Notebook, preprint, arXiv:2608.29219 (2026), Answer 3.2 with Lemma 3.4 and Theorem 3.3 (van der Waall-de Nijs, Theorem 1.15): for p = 2, G = SmallGroup(256, 511) has derived subgroup the Hall-Senior group 32/40, which is the Frattini subgroup of no finite 2-group; the Kourovka Notebook (21st issue) marks 16.11 solved (No, not always), https://arxiv.org/abs/2608.29219.

Uses 1

Statement usesDerived subgroups of p-groups as Frattini subgroups (Kourovka 16.11)

1 fully proved1 ready to prove

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