THE $\boldsymbol {(2,3)}$ -GENERATION OF THE FINITE SIMPLE ODD-DIMENSIONAL ORTHOGONAL GROUPS

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 $\Omega _{2k+1}(q)$ , $k\geq 4$ . As a byproduct, we also obtain $(2,3)$ -generators for $\Omega _{4k}^+(q)$ with $k\geq 3$ and q odd, and for $\Omega _{4k+2}^\pm (q)$ with $k\geq 4$ and $q\equiv \pm 1~ \mathrm {(mod~ 4)}$ .