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14.49 (1999)

Solved

Is $SL_3(\mathbb{Z})$ a factor group of the modular group $PSL_2(\mathbb{Z})$? Since the latter is isomorphic to the free product of two cyclic groups of orders 2 and 3, the question asks if $SL_3(\mathbb{Z})$ can be generated by two elements of orders 2 and 3.

Contributor: M. Conder

Progress

No, it is not (M. C. Tamburini, P. Zucca, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 11, no. 1 (2000), 5–7; Ya. N. Nuzhin, Math. Notes, 70, no. 1–2 (2001), 71–78). M. Conder has shown, however, that $SL_3(\mathbb{Z})$ has a subgroup of index 57 that is a factor group of the modular group $PSL_2(\mathbb{Z})$. Also it has been shown in (M. C. Tamburini, J. S. Wilson, N. Gavioli, J. Algebra, 168 (1994), 353–370) that $SL_d(\mathbb{Z})$ is a factor group of the modular group $PSL_2(\mathbb{Z})$ for all $d \geqslant 28$.

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