Free Burnside groups

AbbreviationFree Burnside group#✓fully proved

For natural numbers \(m\) and \(n\), the free Burnside group \(B(m, n)\) is the quotient of the free group \(F_m\) on \(m\) generators by the normal subgroup generated by all \(n\)-th powers \(w^n\) with \(w \in F_m\).

abbrev FreeBurnsideGroup (m n : ℕ) : Type :=
  FreeGroup (Fin m) ⧸ Subgroup.normalClosure (Set.range fun w : FreeGroup (Fin m) => w ^ n)

Used by 1

Used byMazurov's fixed-point question for free Burnside groups (Kourovka 17.70)

DefinitionMazurov's fixed-point question for free Burnside groups (Kourovka 17.70)#✓fully proved

The assertion that for all primes \(q\) and \(p\) such that the free Burnside group \(B(q, p)\) of rank \(q\) and exponent \(p\) is infinite, the automorphism of \(B(q, p)\) induced by the cyclic permutation \(x_i \mapsto x_{i+1}\) (indices mod \(q\)) of its free generators \(x_0, \dots, x_{q-1}\) fixes some nontrivial element of \(B(q, p)\).

def cyclicAutomorphismFixesNontrivial : Prop :=
  ∀ q p : ℕ, q.Prime → p.Prime → Infinite (FreeBurnsideGroup q p) →
    ∃ w : FreeGroup (Fin q), (w : FreeBurnsideGroup q p) ≠ 1 ∧
      ((FreeGroup.map (finRotate q) w : FreeGroup (Fin q)) : FreeBurnsideGroup q p) = w

Uses 1 · Used by 1

Statement usesFree Burnside group

Used byMazurov's fixed-point question for free Burnside groups has a negative answer

TheoremMazurov's fixed-point question for free Burnside groups has a negative answer#●ready to prove

There exist primes \(q\) and \(p\) such that the free Burnside group \(B(q, p)\) of rank \(q\) and exponent \(p\) is infinite and the automorphism of \(B(q, p)\) induced by the cyclic permutation \(x_i \mapsto x_{i+1}\) (indices mod \(q\)) of its free generators \(x_0, \dots, x_{q-1}\) fixes no nontrivial element of \(B(q, p)\).

public theorem_wanted cyclicAutomorphismFixesNontrivial_refuted :
    ¬ cyclicAutomorphismFixesNontrivial

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 17.70 (V. D. Mazurov); resolution: V. S. Atabekyan, Mazurov's question on fixed points of automorphisms of free Burnside groups, preprint, arXiv:2609.33941 (2026), Theorem 1.1 (for every odd m ≥ 3 there is q_m such that for every prime q > q_m the automorphism of B(m, q) of order m cyclically permuting the free generators has no nontrivial fixed points; B(m, n) is infinite for all odd n ≥ 665 and m > 1 by Adian's theorem, as recalled there); the Kourovka Notebook (21st issue) marks 17.70 solved (No, it does not for all q ≥ 3 and all sufficiently large p), https://arxiv.org/abs/2609.33941.

Uses 1

Statement usesMazurov's fixed-point question for free Burnside groups (Kourovka 17.70)

2 fully proved1 ready to prove

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