The \((2,3)\)-generation of the finite simple odd-dimensional orthogonal groups (Q6622308)

A group is said to be \((2,3)\)-generated if it is generated by an element of order \(2\) and an element of order \(3\). Other than \(\mathrm{PSp}(2^m)\), \(\mathrm{PSp}(3^m)\) and a finite number of undetermined groups, the finite simple classical groups, defined over the finite field of order \(q\) are \((2, 3)\) generated. Many authors have given constructive proofs of this phenomenon (i.e. by explicitly producing the generators).\N\NThe current paper considers the finite orthogonal groups of rank \(\geq 9\) defined over a finite field of odd characteristic and proves that the groups (1) \(\Omega_{2k+1}(q)\) for \(k\geq 4\), (2) \(\Omega_{4k}^{+}(q)\) for \(k\geq 3\), (3) \(\Omega_{4k+2}^{+}(q)\) for \(k\geq 4\) and \(q\equiv 1\pmod{3}\), and (4) \(\Omega_{4k+2}^{-}(q)\) for \(k\geq 4\) and \(q\equiv 3\pmod{4}\) are \((2,3)\)-generated.