The problem of classifying finite projective planes of order n with an automorphism group G and a point orbit O on which G acts two-transitively is investigated in considerable detail, under the assumption that O has length at last n. Combining old and new results a rather satisfying classification is obtained, even though some cases for orbit lengths n and n + 1 remain unsolved. (2000): 51E15, 20B25.
Mathematics Subject Classification
It is shown that, if scriptD is a nontrivial 2‐(v,k,λ) symmetric design, with (k,λ)=1, admitting a flag‐transitive automorphism group G of affine type, then v=pd, p an odd prime, and G is a point‐primitive, block‐primitive subgroup of AΓL1(pd). Moreover, O(G) acts flag‐transitively, point‐primitively on scriptD, and scriptD is isomorphic to the development of a difference set whose parameters and structure are also provided.
It is shown that a projective plane of odd order, with a collineation group acting primitively on the points of an invariant oval, must be desarguesian. Moreover, the group is actually doubly transitive, with only one exception. The main tool in the proof is that a collineation group leaving invariant an oval in a projective plane of odd order has 2‐rank at most three.
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