On Some Maximal Subgroups of Unitary Groups

Abstract

The maximality of certain symplectic subgroups of unitary groups P SU 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.

The geometry of some two-character sets

“…The polarity ρ induces an involutory collineation of Q + (5,4) interchanging O 1 with O 2 . This way we see a group 2 : 3A 6 acting on points of Q + (5,4). Let Y be a point of π 1 , and let Y ⊥ be the polar hyperplane of Y with respect to the orthogonal polarity of Q + (5,4).…”

“…Consider a plane π and a point P in PG (3,4) such that P does not lie on π. Let O be a hyperoval in π (i.e., a set of six points no three of which are collinear; here it is just a conic together with its nucleus) and let O * be the dual hyperoval, i.e., the set of lines of π avoiding O (and indeed forming a hyperoval in the dual of π).…”

“…Let O be a hyperoval in π (i.e., a set of six points no three of which are collinear; here it is just a conic together with its nucleus) and let O * be the dual hyperoval, i.e., the set of lines of π avoiding O (and indeed forming a hyperoval in the dual of π). It is well known (see for example [17]) that the stabilizer of P and O is a group isomorphic to 3A 6 (the triple cover of A 6 ; it is a central perfect extension of A 6 with a group of order 3) and that there is a polarity ρ of PG (3,4) interchanging the six lines P X, X ∈ O, and the six lines of O * . Under the Klein correspondence between the lines of PG (3,4) and the points of the Klein quadric Q + (5, 4), the dual hyperoval O * and the lines P X, where X ranges over O, are represented as two hyperovals O 1 and O 2 embedded in two disjoint totally singular planes, say π 1 and π 2 , respectively.…”

“…It is well known (see for example [17]) that the stabilizer of P and O is a group isomorphic to 3A 6 (the triple cover of A 6 ; it is a central perfect extension of A 6 with a group of order 3) and that there is a polarity ρ of PG (3,4) interchanging the six lines P X, X ∈ O, and the six lines of O * . Under the Klein correspondence between the lines of PG (3,4) and the points of the Klein quadric Q + (5, 4), the dual hyperoval O * and the lines P X, where X ranges over O, are represented as two hyperovals O 1 and O 2 embedded in two disjoint totally singular planes, say π 1 and π 2 , respectively. The polarity ρ induces an involutory collineation of Q + (5,4) interchanging O 1 with O 2 .…”

“…Under the Klein correspondence between the lines of PG (3,4) and the points of the Klein quadric Q + (5, 4), the dual hyperoval O * and the lines P X, where X ranges over O, are represented as two hyperovals O 1 and O 2 embedded in two disjoint totally singular planes, say π 1 and π 2 , respectively. The polarity ρ induces an involutory collineation of Q + (5,4) interchanging O 1 with O 2 . This way we see a group 2 : 3A 6 acting on points of Q + (5,4).…”