On 2-transitive Collineation Groups of Finite Projective Spaces

In 1961, A. Wagner proposed the problem of determining all the subgroups of PΓL(n> q) which are 2-transitive on the points of the projective space PGin — l,q), where n ^ 3. The only known groups with this property are: those containing PSL^n, q), and subgroups of PSL(4, 2) isomorphic to A7. It seems unlikely that there are others, Wagner proved that this is the case when n ^ 5. In unpublished work, D. G. Higman handled the cases n — 6, 7. We will inch up to n S 9. Our result is that nothing surprising happens. The same is true if n = r + 1 for a prime divisor r of q — 1. One of Wagner's results is that it suffices to only consider subgroups of PGL{n> q). Once this is done, it becomes simpler to view the problem as one concerning linear groups: find all those subgroups G of GL(n, q) which are 2-transitive on the 1-spaces of the underlying vector space V. Our approach is based primarily on three facts. (1) Wagner showed that the global stabilizer in G of any 3-space of V induces at least SL(3, q) on that 3-space. (2) Unless G ̂ SL(n, q) or n = 4, q = 2, and G ~ A7, no nontrivial element of G can fix every 1-space of some w-2-space of V. (3) G ̂ SL(n> q) if \G\ is divisible by a prime which is a primitive divisor of q — 1 for a suitable m ^ n — 2.