On the (2,3)-generation of the finite symplectic groups

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

“…M. A. Pellegrini and M. C. Tamburini Bellani [10] PROOF. By condition (ii), H is a subgroup of Ω n (q).…”

“…Suppose now q = p f with f ≥ 2, and let N(q) be the number of elements b ∈ F * q such that F p [b] F q . By [12], we have N(q) ≤ p(p f /2 − 1)/(p − 1), and hence it suffices to check when (p f − 1)/2 − p(p f /2 − 1)/(p − 1) > 4. This condition is fulfilled unless q = 3 2 .…”

“…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.…”

“…(i) Ω 2k+1 (q) with k ≥ 4; (ii) Ω + 4k (q) with k ≥ 3; (iii) Ω + 4k+2 (q) with k ≥ 4 and q ≡ 1 (mod 4); (iv) Ω − 4k+2 (q) with k ≥ 4 and q ≡ 3 (mod 4). We recall that the (2, 3)-generation of Ω 5 (q) PSp 4 (q), when gcd(q, 6) = 1, was proved in [2] (see also [12]). Notice that the groups Ω 5 (3 f ) are not (2, 3)-generated, but they are (2, 5)-generated (see [4]).…”