On some maximal subgroups of unitary groups Antonio Cossidente Dipartimento di Matematica - Universit`a della Basilicata I-85100 Potenza - Italy cossidente@unibas.it Oliver H. King Department of Mathematics - University of Newcastle Newcastle Upon Tyne NE1 7RU, U.K. O.H.King@ncl.ac.uk Abstract The maximality of certain symplectic subgroups of unitary groups PSU n (K), n ≥ 4, (K any field admitting a non–trivial involutory automorphism) belonging to the class C 5 of Aschbacher is proved. Furthermore some related geometry in the case n = 4 and K finite is investigated. Mathematics Subject Classification (2002): 20G40, 20G28 1 Introduction The seminal contibution to the classification of the maximal subgroups of the finite classical groups was Aschbacher’s theorem [3]. Aschbacher defines eight “geometric” classes C 1 ..., C 8 , of subgroups of the finite classical groups and proves that a maximal subgroup either belongs to one of these classes or has a non–abelian simple group as its generalized Fitting subgroup. In their book [12], P.B. Kleidman and M.W. Liebeck have identified the members of the eight classes for modules with dimension greater than 12, 1