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line-transitive point-imprimitive linear spaces with Fang-Li parameter gcd（k, r） at most ten

Frontiers of Mathematics in China 2012年第6期7卷 1095-1112页

作者： Haiyan GUAN Delu TIAN Shenglin ZHOU Department of Mathematics South China University of Technology Guangzhou 510640China

This paper is a further contribution to the classification of line- transitive finite linear spaces. We prove that if y is a non-trivial finite linear space such that the Fang-Li parameter gcd（k, r） is 9 or 10, and the group G ≤ Aut（Y） is line-transitive and point-imprimitive, then y is the Desarguesian projective plane PG（2, 9）.

关键词： linear space line-transitive point-imprimitive

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Finite line-transitive linear Spaces:Theory and Search Strategies

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Acta Mathematica Sinica,English Series 2009年第9期25卷 1399-1436页

作者： Anton BETTEN Anne DELANDTSHEER Maska LAW Alice C. NIEMEYER Cheryl E. PRAEGER Shenglin ZHOU Colorado State University Department of Mathematics Fort Collins CO 80523 U.S.A. Universite Libre de Bruxelles Facultd des Sciences appliquges CP165J11 av. F.D. Roosevelt 50 B 1050 Brussels Belgium The University of Western Australia School of Mathematics and Statistics 35 Stirling Highway Crawley 6009 Australia Department of Mathematics South China University of Technology Guangzhou 510641 China

The paper summarises existing theory and classifications for finite line-transitive linear spaces, develops the theory further, and organises it in a way that enables its effective application. The starting point is a theorem of Camina and the fifth author that identifies three kinds of line-transitive automorphism groups of linear spaces. In two of these cases the group may be imprimitive on points, that is, the group leaves invariant a nontrivial partition of the point set. In the first of these cases the group is almost simple with point-transitive simple socle, and may or may not be point-primitive, while in the second case the group has a non-trivial point-intransitive normal subgroup and hence is definitely point-imprimitive. The theory presented here focuses on point-imprimitive groups. As a non-trivial application a classification is given of the point-imprimitive, line-transitive groups, and the corresponding linear spaces, for which the greatest common divisor gcd（k, v - 1） ≤ 8, where v is the number of points, and k is the line size. Motivation for this classification comes from a result of Weidong Fang and Huffing Li in 1993, that there are only finitely many non-trivial point-imprimitive, linetransitive linear spaces for a given value of gcd（k, v - 1）. The classification strengthens the classification by Camina and Mischke under the much stronger restriction k ≤ 8： no additional examples arise. The paper provides the backbone for future computer-based classifications of point-imprimitive, line- transitive linear spaces with small parameters. Several suggestions for further investigations are made.

关键词： linear space line-transitive automorphism group combinatorial design combinatorial search algorithm

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Solvable line-transitive automorphism groups of finite linear spaces

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Science China Mathematics 2000年第10期43卷 1009-1013页

作者：刘伟俊李慧陵 Department of Mathematics Physics & Mechanics Changsha Railway University Changsha 410075 China Department of Mathematics Zhejiang University Hangzhou 310027 China

Let S be a finite linear space, and let G be a group of automorphisms of S. If G is soluble and line-transitive, then for a given k but a finite number of pairs of ( S, G), S has v= pn points and G≤AΓ L(1,pn).

关键词： linear space line-transitive automorphism solvable group.

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