p-groups with special maximal subgroups

DefinitionSpecial group#✓fully proved

A group \(H\) is special when its center, its commutator subgroup and its Frattini subgroup coincide: \(Z(H) = [H, H] = \Phi(H)\).

def IsSpecialGroup (H : Type*) [Group H] : Prop :=
  Subgroup.center H = commutator H ∧ commutator H = frattini H

Used by 1

Used byKourovka Problem 16.13, strong form (statement)

DefinitionKourovka Problem 16.13, strong form (statement)#✓fully proved

The statement that for every prime \(p\) and every natural number \(n\) there is a finite \(p\)-group \(G\) of order greater than \(n\) all of whose maximal subgroups \(H\) are special, that is, have center \(Z(H)\), commutator subgroup \([H, H]\) and Frattini subgroup \(\Phi(H)\) all equal. This is the form in which the problem was resolved; the problem as posed asks only for one finite \(p\)-group all of whose maximal subgroups are special.

def largePGroupsWithSpecialMaximalSubgroups : Prop :=
  ∀ p : ℕ, p.Prime → ∀ n : ℕ, ∃ (G : Type) (_ : Group G) (_ : Finite G),
    IsPGroup p G ∧ n < Nat.card G ∧ ∀ H : Subgroup G, IsCoatom H → IsSpecialGroup H

Uses 1 · Used by 1

Statement usesSpecial group

Used byArbitrarily large \(p\)-groups all of whose maximal subgroups are special

TheoremArbitrarily large \(p\)-groups all of whose maximal subgroups are special#●ready to prove

For every prime \(p\) and every natural number \(n\) there is a finite \(p\)-group \(G\) of order greater than \(n\) such that every maximal subgroup \(H\) of \(G\) is special, that is, its center \(Z(H)\), its commutator subgroup \([H, H]\) and its Frattini subgroup \(\Phi(H)\) coincide. This answers Kourovka Problem 16.13 positively.

The Kourovka Archive records this answer, citing Cossey (2014); for \(p = 2\) a Sylow \(2\)-subgroup of \(L_3(4)\) is another example (Zenkov).

public theorem_wanted largePGroupsWithSpecialMaximalSubgroups_holds :
    largePGroupsWithSpecialMaximalSubgroups

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 16.13 (Ya. G. Berkovich); resolution: J. Cossey, Bull. Austral. Math. Soc. 89 (2014), 415-419, Kourovka Archive, Problem 16.13 (for all primes, groups of arbitrarily large size; for p = 2 also a Sylow 2-subgroup of L_3(4), V. I. Zenkov, 2014).

Uses 1

Statement usesKourovka Problem 16.13, strong form (statement)

2 fully proved1 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.