Generalized torsion

DefinitionGeneralized torsion element (Kourovka 16.49)#✓fully proved

An element \(h\) of a group \(H\) is a generalized torsion element when \(h \ne 1\) and some nonempty product of conjugates of \(h\) is trivial, that is, \(x_1^{-1} h x_1 \cdots x_k^{-1} h x_k = 1\) for some \(k \ge 1\) and some \(x_1, \dots, x_k \in H\).

def IsGeneralizedTorsion {H : Type*} [Group H] (h : H) : Prop :=
  h ≠ 1 ∧ ∃ l : List H, l ≠ [] ∧ (l.map fun x => x⁻¹ * h * x).prod = 1

Used by 1

Used byKopytov–Medvedev question on generalized torsion in free products (Kourovka 16.49)

DefinitionKopytov–Medvedev question on generalized torsion in free products (Kourovka 16.49)#✓fully proved

The statement that for every family of groups \((G_i)_{i \in I}\) none of which contains a generalized torsion element (an element \(h \ne 1\) such that some nonempty product of conjugates of \(h\) is trivial), the free product of the family contains no generalized torsion element either.

def freeProductNoGeneralizedTorsion : Prop :=
  ∀ (ι : Type) (G : ι → Type) [∀ i, Group (G i)],
    (∀ i, ∀ g : G i, ¬ IsGeneralizedTorsion g) → ∀ g : Monoid.CoprodI G, ¬ IsGeneralizedTorsion g

Uses 1 · Used by 1

Statement usesGeneralized torsion element (Kourovka 16.49)

Used byFree products of groups without generalized torsion have no generalized torsion

TheoremFree products of groups without generalized torsion have no generalized torsion#●ready to prove

For every family of groups \((G_i)_{i \in I}\) none of which contains a generalized torsion element (an element \(h \ne 1\) such that some nonempty product of conjugates of \(h\) is trivial), the free product of the family contains no generalized torsion element.

The Kourovka Archive answers 16.49 "Yes, it is true", citing Ito, Motegi and Teragaito (2019), whose Theorem 1.5 shows that a generalized torsion element of a free product \(G_1 \ast \cdots \ast G_n\) of torsion-free groups is conjugate to a generalized torsion element of some factor.

public theorem_wanted freeProductNoGeneralizedTorsion_holds : freeProductNoGeneralizedTorsion

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 16.49 (V. M. Kopytov, N. Ya. Medvedev); resolution: T. Ito, K. Motegi and M. Teragaito, Generalized torsion and decomposition of 3-manifolds, Proc. Amer. Math. Soc. 147 (2019), no. 11, 4999-5008, arXiv:1811.07532, Theorem 1.5 (a generalized torsion element of a free product G_1 ∗ ⋯ ∗ G_n of torsion-free groups is conjugate to a generalized torsion element of some factor); Kourovka Archive, Problem 16.49 (Yes; moreover the generalized torsion in a free product of torsion-free groups is conjugate to a generalized torsion of one of its factor groups), https://arxiv.org/abs/1811.07532.

Uses 1

Statement usesKopytov–Medvedev question on generalized torsion in free products (Kourovka 16.49)

DefinitionGeneralized torsion elements (Kourovka 16.48)#✓fully proved

An element \(h\) of a group \(H\) is a generalized torsion element if \(h \ne 1\) and there are \(n \ge 1\) elements \(x_1, \dots, x_n \in H\), not necessarily distinct, such that \((x_1^{-1} h x_1)(x_2^{-1} h x_2) \cdots (x_n^{-1} h x_n) = 1\).

def IsGeneralizedTorsion {H : Type*} [Group H] (h : H) : Prop :=
  h ≠ 1 ∧ ∃ l : List H, l ≠ [] ∧ (l.map fun x => x⁻¹ * h * x).prod = 1

Used by 1

Used byThe Kopytov–Medvedev question (Kourovka 16.48)

DefinitionRight orders on a group (Kourovka 16.48)#✓fully proved

The right orders of a group \(G\) are the binary relations \(\preceq\) on \(G\) that are linear orders (reflexive, transitive, antisymmetric and total) and are invariant under right multiplication: \(a \preceq b\) implies \(ac \preceq bc\) for all \(a, b, c \in G\).

def rightOrders (G : Type*) [Group G] : Set (G → G → Prop) :=
  {r | IsLinearOrder G r ∧ ∀ a b c : G, r a b → r (a * c) (b * c)}

Used by 1

Used byThe Kopytov–Medvedev question (Kourovka 16.48)

DefinitionThe Kopytov–Medvedev question (Kourovka 16.48)#✓fully proved

This is the proposition that every group \(H\) with no generalized torsion element is right-orderable, that is, admits a right order.

def noGeneralizedTorsionRightOrderable : Prop :=
  ∀ (H : Type) [Group H], (∀ h : H, ¬ IsGeneralizedTorsion h) → (rightOrders H).Nonempty

Uses 2 · Used by 1

Statement usesGeneralized torsion elements (Kourovka 16.48) · Right orders on a group (Kourovka 16.48)

Used bySome group without generalized torsion is not right-orderable (Kourovka 16.48)

TheoremSome group without generalized torsion is not right-orderable (Kourovka 16.48)#●ready to prove

The Kopytov–Medvedev proposition is false: there is a group \(H\) with no generalized torsion element that admits no right order.

Resolved false: Cai and Clay give a condition ensuring that an amalgam is generalized torsion-free and use it to construct a group that is generalized torsion-free but not left-orderable (their Theorem 1.3, proved in Section 7), hence not right-orderable; the Kourovka Notebook (21st issue) answers 16.48 "No, not every".

public theorem_wanted noGeneralizedTorsionRightOrderable_refuted :
    ¬ noGeneralizedTorsionRightOrderable

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 16.48 (V. M. Kopytov, N. Ya. Medvedev); resolution: T. W. Cai and A. Clay, Generalized torsion in amalgams, preprint, arXiv:2504.08084 (2025), Theorem 1.3 (there exists a group which is generalized torsion-free and not left-orderable; Definition 7.1 and Theorem 7.2); the Kourovka Notebook (21st issue) marks 16.48 solved (No, not every), https://arxiv.org/abs/2504.08084.

Uses 1

Statement usesThe Kopytov–Medvedev question (Kourovka 16.48)

5 fully proved2 ready to prove

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