Coclass and rank of p-groups

DefinitionMinimal number of generators of a subgroup#✓fully proved

For a subgroup \(H\) of a group \(G\), the minimal number of generators of \(H\) is the least natural number \(n\) such that \(H\) is generated by some finite set of \(n\) elements of \(G\); it is \(0\) when \(H\) is not finitely generated.

noncomputable def minGenerators {G : Type*} [Group G] (H : Subgroup G) : ℕ :=
  sInf {n | ∃ s : Finset G, s.card = n ∧ Subgroup.closure (s : Set G) = H}

Used by 1

Used byRank of a group

DefinitionRank of a group#✓fully proved

The rank of a group \(G\) is the supremum, in the natural numbers, of the minimal numbers of generators of the subgroups of \(G\), where an unbounded set of natural numbers has supremum \(0\). For a finite group \(G\) it is the largest minimal number of generators of a subgroup of \(G\).

noncomputable def groupRank (G : Type*) [Group G] : ℕ :=
  sSup (Set.range (minGenerators (G := G)))

Uses 1 · Used by 1

Statement usesMinimal number of generators of a subgroup

Used byRank analogue of the Shepherd-Leedham-Green-McKay theorem (Kourovka 16.103)

DefinitionCyclic lower central factors from the second on#✓fully proved

Let \(\gamma_1(P) = P\) and \(\gamma_{i+1}(P) = [\gamma_i(P), P]\) be the lower central series of a group \(P\). The group \(P\) has cyclic lower central factors from the second on if for every \(i \ge 2\) there is an element \(g \in \gamma_i(P)\) such that every \(x \in \gamma_i(P)\) satisfies \(x g^{-k} \in \gamma_{i+1}(P)\) for some integer \(k\), that is, the factor \(\gamma_i(P)/\gamma_{i+1}(P)\) is cyclic.

def HasCyclicLowerCentralFactorsFromSecond (P : Type*) [Group P] : Prop :=
  ∀ j : ℕ, 1 ≤ j → ∃ g ∈ (⊤ : Subgroup P).lowerCentralSeries j,
    ∀ x ∈ (⊤ : Subgroup P).lowerCentralSeries j,
      ∃ k : ℤ, x * (g ^ k)⁻¹ ∈ (⊤ : Subgroup P).lowerCentralSeries (j + 1)

Used by 1

Used byRank analogue of the Shepherd-Leedham-Green-McKay theorem (Kourovka 16.103)

DefinitionRank analogue of the Shepherd-Leedham-Green-McKay theorem (Kourovka 16.103)#✓fully proved

The statement that for some prime \(p\) there is a natural number \(r\) such that every finite \(p\)-group \(P\) generated by two elements whose lower central factors \(\gamma_i(P)/\gamma_{i+1}(P)\) are cyclic for all \(i \ge 2\) has a normal subgroup \(N\) of nilpotency class at most \(2\) with \(P/N\) of rank at most \(r\). Here \(\gamma_1(P) = P\) and \(\gamma_{i+1}(P) = [\gamma_i(P), P]\), and the rank of a finite group is the largest minimal number of generators of its subgroups.

Kourovka 16.103 (E. I. Khukhro) asks for a bound \(r(p)\) for each prime \(p\); this form asks for one prime, so its negation says that no prime has such a bound.

def rankAnalogueMaximalClass : Prop :=
  ∃ p : ℕ, p.Prime ∧ ∃ r : ℕ, ∀ (P : Type) [Group P] [Finite P], IsPGroup p P →
    (∃ x y : P, Subgroup.closure ({x, y} : Set P) = ⊤) → HasCyclicLowerCentralFactorsFromSecond P →
      ∃ N : Subgroup P, ∃ _ : N.Normal, N.lowerCentralSeries 2 = ⊥ ∧ groupRank (P ⧸ N) ≤ r

Uses 2 · Used by 1

Statement usesCyclic lower central factors from the second on · Rank of a group

Used byNo rank analogue of the Shepherd-Leedham-Green-McKay theorem (Kourovka 16.103)

TheoremNo rank analogue of the Shepherd-Leedham-Green-McKay theorem (Kourovka 16.103)#●ready to prove

For every prime \(p\) and every natural number \(r\) there is a finite \(p\)-group \(P\) generated by two elements, whose lower central factors \(\gamma_i(P)/\gamma_{i+1}(P)\) are cyclic for all \(i \ge 2\), such that for every normal subgroup \(N\) of \(P\) of nilpotency class at most \(2\) the quotient \(P/N\) has rank greater than \(r\). Here \(\gamma_1(P) = P\) and \(\gamma_{i+1}(P) = [\gamma_i(P), P]\), and the rank of a finite group is the largest minimal number of generators of its subgroups. That is, the statement of Kourovka Problem 16.103 fails for every prime.

Khukhro's groups even have no normal subgroup of bounded derived length with quotient of bounded rank; the Kourovka Archive answers 16.103 "No".

public theorem_wanted rankAnalogueMaximalClass_refuted : ¬ rankAnalogueMaximalClass

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 16.103 (E. I. Khukhro); resolution: E. I. Khukhro, Counterexamples to a rank analogue of the Shepherd-Leedham-Green-McKay theorem on finite p-groups of maximal class, Siberian Math. J. 54 (2013), no. 1, 174-184, Theorem (no functions r(p), l(p) give every such group a normal subgroup of derived length at most l(p) with quotient of rank at most r(p)), with examples for every fixed prime (Section 4); Kourovka Archive, Problem 16.103 (answer: No), https://eprints.maths.manchester.ac.uk/1937/1/.

Uses 1

Statement usesRank analogue of the Shepherd-Leedham-Green-McKay theorem (Kourovka 16.103)

4 fully proved1 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.