The (2, 3)-generation of the finite 8-dimensional orthogonal groups

Abstract

We construct ( 2 , 3 ) (2,3) -generators for the finite 8-dimensional orthogonal groups, proving the following results: the groups Ω 8 + ⁢ ( q ) \Omega_{8}^{+}(q) and P ⁢ Ω 8 + ⁢ ( q ) \mathrm{P}\Omega_{8}^{+}(q) are ( 2 , 3 ) (2,3) -generated if and only if q ≥ 4 q\geq 4 ; the groups Ω 8 - ⁢ ( q ) \Omega_{8}^{-}(q) and P ⁢ Ω 8 - ⁢ ( q ) \mathrm{P}\Omega_{8}^{-}(q) are ( 2 , 3 ) (2,3) -generated for all q ≥ 2 q\geq 2 .

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

“…LEMMA 5.2. The group K |S 9 is neither monomial nor contained in any maximal subgroup PSL 2 (8), PSL 2 (17), Alt (10), Sym (10), Sym (11) in class S of Ω 9 (q). PROOF.…”