(2,3)-generation of groups

Definition\((2,3)\)-generation of \(\mathrm{SL}_3(\mathbb{Z})\) (Kourovka 14.49)#✓fully proved

The statement that there are \(a, b \in \mathrm{SL}_3(\mathbb{Z})\) such that \(a\) has order \(2\), \(b\) has order \(3\), and \(a\) and \(b\) generate \(\mathrm{SL}_3(\mathbb{Z})\).

Kourovka Notebook Problem 14.49 (M. Conder) asks whether \(\mathrm{SL}_3(\mathbb{Z})\) is a quotient of the modular group \(\mathrm{PSL}_2(\mathbb{Z}) \cong C_2 \ast C_3\); since \(\mathrm{SL}_3(\mathbb{Z})\) is not cyclic, this is equivalent to the statement above.

def sl3IntGeneratedByOrderTwoAndThree : Prop :=
  ∃ a b : Matrix.SpecialLinearGroup (Fin 3) ℤ, orderOf a = 2 ∧ orderOf b = 3 ∧
    Subgroup.closure {a, b} = ⊤

Used by 1

Used by\(\mathrm{SL}_3(\mathbb{Z})\) is not \((2,3)\)-generated

Theorem\(\mathrm{SL}_3(\mathbb{Z})\) is not \((2,3)\)-generated#●ready to prove

There are no \(a, b \in \mathrm{SL}_3(\mathbb{Z})\) such that \(a\) has order \(2\), \(b\) has order \(3\), and \(a\) and \(b\) generate \(\mathrm{SL}_3(\mathbb{Z})\). Equivalently, \(\mathrm{SL}_3(\mathbb{Z})\) is not a quotient of the modular group \(\mathrm{PSL}_2(\mathbb{Z}) \cong C_2 \ast C_3\).

This is the negative answer to Kourovka Notebook Problem 14.49 (M. Conder), due to Tamburini and Zucca, and Nuzhin.

public theorem_wanted sl3IntGeneratedByOrderTwoAndThree_refuted :
    ¬ sl3IntGeneratedByOrderTwoAndThree

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 14.49 (M. Conder); resolution: M. C. Tamburini and P. Zucca, Rend. Lincei Mat. Appl. (9) 11 (2000), no. 1, 5-7; Ya. N. Nuzhin, Math. Notes 70 (2001), no. 1-2, 71-78, Kourovka Archive, Problem 14.49 (answer: No, it is not; SL_d(ℤ) is such a quotient for d ≥ 28).

Uses 1

Statement uses\((2,3)\)-generation of \(\mathrm{SL}_3(\mathbb{Z})\) (Kourovka 14.49)

1 fully proved1 ready to prove

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