Generation of \(SL(n,\mathbb{Z})\) by a Jordan unipotent matrix and its transpose
Let \(SL(n,\mathbb{Z})\) denote the group formed by all \(n\times n\) matrices of determinant 1 with entries in \(\mathbb{Z}\). Then \(SL(n,\mathbb{Z})\) contains the \(n\times n\) Jordan unipotent block matrix \(x\) whose only eigenvalue is 1, and the transpose \(y\) of \(x\). For \(n\geq 2\) and \(n\neq 4\) the authors show that \(x\) and \(y\) generate \(SL(n,\mathbb{Z})\). In the case \(n = 4\), the matrices \(x\) and \(y\) generate a subgroup of index 8 in \(SL(n,\mathbb{Z})\).
scientific article; zbMATH DE number 167833