Peter B. Kleidman

The maximal subgroups of the finite 8-dimensional orthogonal groups PΩ8+(q) and of their automorphism groups

During this century, and even before, a substantial amount of work has been devoted to finding the maximal subgroups of finite simple groups and their automorphism groups. Some of the earliest published results of this nature appear in Wiman [42] and Moore [33], where the maximal subgroups of the groups L,(q) = PSL,(q) are determined for all q. Then Mitchell [31, 321 and Hartley [15] found the maximal subgroups of L,(q), U,(q) = PSU,(q), and also PSpJq) for odd q. Several decades later, analogous results appeared for various other low-dimensional classical groups, including P,!+,(q) with q even, L,(q), U,(q), and L,(q). (The bibliographies of [lo] and [43] serve as good sources of reference.) Recently Aschbacher [2] made a significant contribution to the solution of the problem of finding the maximal subgroups of any group whose socle is a classical simple group. The main theorem of [2] says the following. Let Go be a linite classical simple group with natural projective module V, and let G be a group with socle G, (i.e., G, a G9 Aut(G,)). Assume that if GOr PQ,t(q) then G does not contain a triality automorphism of G,. If M is a maximal subgroup of G not containing G,, then one of the following holds: IS on ;“’ A4 a known group with a well-described projective action

New families of ovoids in O+8

We construct four new infinite families of ovoids in the 8-dimensional orthogonal geometry Oinf8sup+. We determine the automorphism groups of these ovoids and we show that the two ‘sporadic’ ovoids recently found by Cooperstein [2] and Shult [11] are members of our families.

A Survey of the Maximal Subgroups of the Finite Simple Groups

We survey some recent results on maximal subgroups of the finite simple groups. In particular, we describe progress on several of the problems raised by Aschbacher in [3].

The 2-transitive ovoids

An ovoid 0 in a classical polar space is a set of singular points such that every maximal totally singular subspace contains just one point in Lo. Ovoids are intimately connected with other combinatorial objects, including translation planes, spreads, partial geometries, codes, generalized hexagons, and Kerdock sets (see [S, 9, 163 for example). Interestingly enough, many of the known ovoids are in fact 2-transitive-that is, they admit a 2-transitive automorphism group (the notion of the automorphism group is made precise in (2.3)). For example, the Suzuki groups Sz(q) and the Ree groups *G,(q) act 2-transitively on ovoids in 4-dimensional symplectic geometry and 7-dimensional orthogonal geometry, respectively. Furthermore, the unitary groups PSU,(q) (for suitable prime powers q) and the linear groups PSL,(q3) (with q even) act 2-transitively on ovoids in 7or 8-dimensional orthogonal geometry. The occurrence of such a large number of 2-transitive ovoids suggests that a classification of them is worthwhile, much in the same spirit as Kantor’s classification of the finite linear spaces whose automorphism group acts 2-transitively on points [lo]. The classification of the 2-transitive ovoids appears as our Main Theorem in Section 2. We discover no new ovoids, however we obtain some new results concerning the number of isomorphism classes of 2-transitive ovoids (see Section 2). Our proof relies on the classification of the finite 2-transitive permutation groups, which in turn relies on the recent classification of finite simple groups. WC also draw upon several facts from the modular representation theory of finite groups.

On a theorem of Feit and Tits

Feit and Tits [3] lay the groundwork for determining the smallest degree of a projective representation of a finite extension of a finite simple group G. Provided G is not of Lie type in characteristic 2, they determine precisely when this degree is smaller than the degree of a projective representation of G itself. We complete this project by extending their results to the groups of Lie type in characteristic 2.