On the \((2,3)\)-generation of some classical groups. I
A \((2,3)\)-generated group is one which can be generated by a pair of elements \(x\), \(y\), where \(x\) and \(y\) have orders 2 and 3, respectively. Such a group is then an image of \(\text{PSL}_ 2(\mathbb{Z})\). Using a generalization of a technique from the theory of permutation groups introduced by Wielandt, the authors prove that, if \(R\) is a commutative ring, generated by \(d\) units of finite order, then, for all \(n \geq 12d + 16\), the group generated by the \(n \times n\) elementary matrices, \(E_ n(R)\), is \((2,3)\)-generated. It follows that, for \(n\) sufficiently large, \(\text{SL}_ n(R)\) is \((2,3)\)-generated, where \(R\) is (i) a Euclidean domain, (ii) a semi-local ring or (iii) a Hasse domain in a global field. Similar results are proved for the classical groups \(\text{Sp}_{2n}(R)\), \(\Omega^ +_{2n}(R)\) and \(\text{SU}_{2n}(R)\), where \(R\) is of type (i), (ii) or (iii). The same techniques can be used to prove results of a more abstract group-theoretic nature. Let \(R\) be a finitely generated commutative ring and let \({\mathcal C} = \{E_ n(R) : n > 2\}\). They show that \(\text{PSL}_ 2(\mathbb{Z})\) is residually-\({\mathcal C}\), i.e. the intersection of all normal subgroups \(N\) of \(\text{PSL}_ 2(\mathbb{Z})\), such that \(\text{PSL}_ 2(\mathbb{Z})/N \in {\mathcal C}\), is trivial.
scientific article; zbMATH DE number 645084