Extremely line-primitive automorphism groups of finite linear spaces

摘要

A primitive non-regular permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. This notation was first introduced in the work of Manning in the 1920s. In 1988, Delandtsheer and Doyen conjectured that line-primitivity can imply point-primitivity for the automorphism group of a finite linear space. Let G be an automorphism group of a nontrivial finite regular linear space S. We prove that, if G is extremely line-primitive, then S is a finite projective plane and G is extremely point-primitive. This result supports the Delandtsheer-Doyen conjecture. We then explore projective planes of order n admitting an extremely line-primitive automorphism group and bound the line rank of G with the polynomials related to n. In particular, if n is a prime power, then the two cases that the line rank of G attains the lower bound n + 1 or the upper bound n(2)+n+3/3 are separately investigated.

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