Infinite 2-group quotients of triangle groups

AbbreviationThe (2, 4, 8) triangle group#✓fully proved

The \((2, 4, 8)\) triangle group is the group with presentation \(\langle x, y \mid x^2 = y^4 = (xy)^8 = 1 \rangle\).

abbrev wiegoldGroup : Type :=
  PresentedGroup ({FreeGroup.of 0 ^ 2, FreeGroup.of 1 ^ 4, (FreeGroup.of 0 * FreeGroup.of 1) ^ 8} :
    Set (FreeGroup (Fin 2)))

Used by 1

Used byWiegold's question on 2-group quotients of the (2, 4, 8) triangle group (Kourovka 16.101)

DefinitionWiegold's question on 2-group quotients of the (2, 4, 8) triangle group (Kourovka 16.101)#✓fully proved

The statement that there are an uncountable index set \(I\) and normal subgroups \(N_i\) of \(\Gamma = \langle x, y \mid x^2 = y^4 = (xy)^8 = 1 \rangle\), for \(i \in I\), such that every quotient \(\Gamma / N_i\) is an infinite \(2\)-group (a group in which every element has order a power of \(2\)), and \(\Gamma / N_i \cong \Gamma / N_j\) only when \(i = j\).

def uncountablyManyInfiniteTwoGroupQuotients : Prop :=
  ∃ (ι : Type) (_ : Uncountable ι) (N : ι → Subgroup wiegoldGroup) (_ : ∀ i, (N i).Normal),
    (∀ i, Infinite (wiegoldGroup ⧸ N i) ∧ IsPGroup 2 (wiegoldGroup ⧸ N i)) ∧
      ∀ i j, Nonempty ((wiegoldGroup ⧸ N i) ≃* (wiegoldGroup ⧸ N j)) → i = j

Uses 1 · Used by 1

Statement usesThe (2, 4, 8) triangle group

Used byUncountably many infinite 2-group quotients of the (2, 4, 8) triangle group (Kourovka 16.101)

TheoremUncountably many infinite 2-group quotients of the (2, 4, 8) triangle group (Kourovka 16.101)#●ready to prove

Let \(\Gamma = \langle x, y \mid x^2 = y^4 = (xy)^8 = 1 \rangle\). There are an uncountable index set \(I\) and normal subgroups \(N_i\) of \(\Gamma\), for \(i \in I\), such that every quotient \(\Gamma / N_i\) is an infinite \(2\)-group (a group in which every element has order a power of \(2\)), and \(\Gamma / N_i \cong \Gamma / N_j\) only when \(i = j\). So Wiegold's question has a positive answer.

The Kourovka Archive answers 16.101 "Yes, there do" (Grigorchuk; Minasyan, Olshanskii and Sonkin).

public theorem_wanted uncountablyManyInfiniteTwoGroupQuotients_holds :
    uncountablyManyInfiniteTwoGroupQuotients

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 16.101 (J. Wiegold); resolution: R. I. Grigorchuk, Algebra Discrete Math. 2009, no. 4, 78-96; A. Minasyan, A. Yu. Olshanskii and D. Sonkin, Periodic quotients of hyperbolic and large groups, Groups Geom. Dyn. 3 (2009), no. 3, 423-452, arXiv:0804.3328, Kourovka Archive, Problem 16.101 (answer: Yes, there do); Minasyan, Olshanskii and Sonkin, abstract (continua of pairwise non-isomorphic periodic quotients, with a positive answer to Wiegold's question), https://arxiv.org/abs/0804.3328.

Uses 1

Statement usesWiegold's question on 2-group quotients of the (2, 4, 8) triangle group (Kourovka 16.101)

2 fully proved1 ready to prove

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