Vol. 48, No. 1, 1973

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On 2-transitive collineation groups of finite projective spaces

William M. Kantor

Vol. 48 (1973), No. 1, 119–131

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(n1,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 In unpublished work, D. G. Higman handled the cases n = 6,7.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.

Mathematical Subject Classification 2000
Primary: 20H15
Secondary: 50D30
Received: 6 November 1971
Revised: 13 October 1972
Published: 1 September 1973
William M. Kantor