Transitive projective planes
نویسنده
چکیده
A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the points of a non-Desarguesian projective plane must not contain any components. 1 Background definitions and main results We say that a projective plane is transitive (respectively primitive) if it admits an automorphism group which is transitive (respectively primitive) on points. Kantor [22] has proved that a projective plane P of order x admitting a point-primitive automorphism group G is Desarguesian and G ≥ PSL(3, x), or else x + x + 1 is a prime and G is a regular or Frobenius group of order dividing (x + x+ 1)(x+ 1) or (x + x+ 1)x. Kantor’s result, which depends upon the Classification of Finite Simple Groups, represents the strongest success in the pursuit of a proof to the conjecture mentioned in the abstract. A corollary of Kantor’s result is that a group acts primitively on the points of a projective plane P if and only if it acts primitively on the lines of P. We also know, by a combinatorial argument of Block, that a group acts transitively on the points of a projective plane P if and only if it acts transitively on the lines of P [5]. Our primary result is the following: Theorem A. Suppose that G acts transitively on a projective plane P of order x. Then one of the following cases holds: • P is Desarguesian, G ≥ PSL(3, x) and the action is 2-transitive on points; • G does not contain a component. In particular all minimal normal subgroups of G are elementary abelian. This paper contains results from the author’s PhD thesis. I would like to thank my supervisor, Professor Jan Saxl. Professor Bill Kantor has also given much helpful advice for which I am very grateful.
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