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On the orders of maximal subgroups of the finite classical groups. (English) Zbl 0591.20021
Let $$G_ 0$$ be a simple classical group (that is, a projective special linear, symplectic, unitary or orthogonal group) with natural projective module V of dimension n over GF(q), and let G be a group such that $$G_ 0\triangleleft G\leq Aut(G_ 0)$$. The main result of the paper: if H is any maximal subgroup of G, then one of the following holds: (I) H is a known group (and $$H\cap G_ 0$$ has well-described projective action on V); (II) $$| H| <q^{3n}.$$
The proof of this result uses the classification of finite simple groups and improves the results of B. N. Cooperstein [Isr. J. Math. 30, 213-235 (1978; Zbl 0383.20027)] and W. M. Kantor [J. Algebra 60, 158-168 (1979; Zbl 0422.20033)]. Corresponding result for alternating groups cf. Theorem 6.1 of P. J. Cameron [Bull. Lond. Math. Soc. 13, 1-22 (1981; Zbl 0463.20003)]. Complete lists of maximal subgroups are known for many of the sporadic groups. For the case where $$G_ 0$$ is an exceptional group of Lie type bounds were obtained by the author and J. Saxl [On the orders of maximal subgroups of the finite exceptional groups of Lie type (Preprint, Univ. Cambridge 1984)].