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

Abstract

The complete classification of the finite simple groups that are $(2,3)$-generated is a problem which is still open only for orthogonal groups. Here, we construct $(2,3)$-generators for the finite odd-dimensional orthogonal groups ${\Omega }_{2k+1}(q)$, $k\ge 4$. As a byproduct we also obtain $(2,3)$-generators for ${\Omega }_{4k}^{+}(q)$ with $k\ge 3$ and $q$ odd, and for ${\Omega }_{4k+2}^{\pm }(q)$ with $k\ge 4$ and $q\equiv \pm 1(\mathrm{mod}4)$.