Cartesian powers of free groups

DefinitionTorsion-freeness of abelianized Cartesian powers of free groups (Kourovka 13.18)#✓fully proved

The statement, posed by A. M. Gaglione and D. Spellman, that for every integer \(n \ge 2\) the abelianization of the unrestricted (Cartesian) product \(F_n^{\mathbb{N}}\) of countably many copies of the free group \(F_n\) of rank \(n\) is torsion-free: none of its nontrivial elements has finite order.

def cartesianPowerAbelianizationTorsionFree : Prop :=
  ∀ n : ℕ, 2 ≤ n → ∀ g : Abelianization (ℕ → FreeGroup (Fin n)), g ≠ 1 → ¬ IsOfFinOrder g

Used by 1

Used byAbelianized Cartesian powers of free groups can have torsion

TheoremAbelianized Cartesian powers of free groups can have torsion#●ready to prove

For some integer \(n \ge 2\), the abelianization of the unrestricted (Cartesian) product \(F_n^{\mathbb{N}}\) of countably many copies of the free group \(F_n\) of rank \(n\) contains a nontrivial element of finite order. That is, the torsion-freeness asked for in Kourovka Problem 13.18 fails.

The Kourovka Archive answers 13.18 "No, it may contain elements of order 2" (O. Kharlampovich and A. Myasnikov, 2001).

public theorem_wanted cartesianPowerAbelianizationTorsionFree_refuted :
    ¬ cartesianPowerAbelianizationTorsionFree

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 13.18 (A. M. Gaglione, D. Spellman); resolution: O. Kharlampovich and A. Myasnikov, in: Knots, braids, and mapping class groups (New York, 1998), Amer. Math. Soc., 2001, 77-83, Kourovka Archive, Problem 13.18: No, it may contain elements of order 2.

Uses 1

Statement usesTorsion-freeness of abelianized Cartesian powers of free groups (Kourovka 13.18)

1 fully proved1 ready to prove

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