Antiflag-transitive collineation groups revisited

An antiflag in a projective space is a non-incident point-hyperplane pair. A subgroup G of ΓL(n,q) is antiflag-transitive if it acts transitively on the set of antiflag of PG(n−1,q). In 1979, Cameron and Kantor [2] published a paper determining all antiflagtransitive subgroups of ΓL(n,q). A large part of the motivation was the fact that a group which acts 2-transitively on points is necessarily antiflag-transitive, so that the result included the solution to Wagner’s problem of determining the 2transitive collineation groups. At about the same time, this problem was also settled by Orchel in his London Ph.D. thesis [7]; his approach was also based on antiflag-transitivity, though he did not determine all the antiflag-transitive groups. The techniques used by Cameron and Kantor were largely geometric (involving translation planes and generalized polygons, among other things). Essentially no group theory beyond Sylow’s Theorem and its consequences was used. Shortly afterwards, the classification of finite simple groups was announced (though there were some problems with the proof, which may now have been resolved). Using this classification, Hering [4] showed that it is possible to determine all subgroups of ΓL(n,q) which act transitively on the points of projective space, a much stronger result. However, the Cameron–Kantor theorem has been used in various places: by Kantor [6] in determining collineation groups containing a Singer cycle, and by Abhyankar (see [1] and subsequent papers) in showing that various groups are Galois groups in non-zero characteristic, as well as others [3, 9, 10]. So it perhaps retains its interest. One purpose of this note is to give a self-contained account.