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 PG(n− 1,q), where n ≧ 3. The only
known groups with this property are: those containing PSL(n,q), and subgroups
of PlSL(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 In unpublished work, D. G.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 ≦ 9. Our result is
that nothing surprising happens. The same is true if n = rα+ 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 l-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 l-space of some n-2-space of V . (3) G ≦ SL(n,q) if |G| is
divisible by a prime which is a primitive divisor of qm− 1 for a suitable
m ≦ n − 2.
