On \((2,3)\)-generated groups

M. Vsemirnov

A \((2,3)\)-generated group is a group that can be generated by an involution and an element of order \(3\). The interest in these groups is explained by the well-known fact that the quotients of the modular group \(\mathrm{PSL}_2(\mathbb{Z})\) (except the three "degenerate" cases \(\{ 1 \}\), \(C_2\), \(C_3\)) are precisely the \((2,3)\)-generated groups.

In my talk, I shall survey recent results in that field including

 •  explicit constructions of \((2,3)\)-generators for various matrix groups,
 •  new additions to the list of non-\((2,3)\)-generated finite simple groups.

See also the author's pdf version: