On completely reducible solvable subgroups of GL(n,\(\Delta\) ) (Q1066276)

Let q be a prime power, and let b(n,q) denote the maximal possible order of a solvable, completely reducible subgroup of the general linear group GL(n,q). \textit{T. R. Wolf} has shown that \(b(n,q)\leq q^{(1+\beta)n}/24^{1/3}\) where \(\beta\) lies between 1.2 and 1.25 for all \(q\geq 2\) and \(n\geq 1\) [see Can. J. Math. 34, 1097-1111 (1982; Zbl 0476.20028)]. In the present paper the author proves (Theorem A): \(b(n,q)\leq \gamma_ nq^ n\) for \(n\geq 1\) and \(q\geq 11\) (an added note asserts the result is also true for \(q\geq 8)\) where \(\gamma_ n\) denotes the maximal possible order of a solvable subgroup of the symmetric group \(S_ n\). Since it is known that \(\gamma_ n\leq 24^{(n-1)/3}\) [see \textit{J. D. Dixon}, J. Aust. Math. Soc. 7, 417-424 (1967; Zbl 0153.040)], this theorem improves Wolf's result for \(q\geq 11\). For \(q>13\) the author describes the solvable subgroups of order b(n,q) (Theorem B): These subgroups are conjugate to a wreath product of the form \(\Delta^* wr \Gamma_ n\) if \(n\not\equiv 2\), 10 or 14 (mod 16), and to a direct product \(H\times (\Delta^* wr \Gamma_{n-2})\) otherwise. Here \(\Gamma_ n\) denotes the (unique up to conjugacy) subgroup of order \(\gamma_ n\) in \(S_ n\), \(\Delta^*\) is the multiplicative group of the field of order q, and H is a solvable irreducible subgroup of order \(2(q^ 2-1)\) in GL(2,q).