Point Transitive Finite Linear Spaces

A linear space S is an incidence structure (V, C), where P is a set of points, C is a set of distinguish subsets of P called lines, every line must be incident with at least two points and every pair of distinct points is incident with one line exactly. An automorphism group G of S is a permutation group on P which leaves C invariant. The research about linear spaces and their automorphism groups is an important problem between groups and combinatorial designs. Generally, the characteristic of the action of groups on linear spaces are depicted by the transitivity or primitivity of groups on some special sets, where the sets are the set of points, lines, flags and so on. The problems of classification of non-trivial finite linear spaces are focus on the classification of linear spaces admitting automorphism groups which have some properties. In this thesis, we are going to discuss the classification of non-trivial finite linear spaces with point-transitive automorphism groups. Trying to deal with some special cases for finding general results about linear spaces, will help us to obtain more theory and experience in this field.First of all, we present a survey of the backgrounds and modern developments of design and group theory, describe the main research work of this thesis, and then give some preliminary results on finite linear spaces and groups, which are used in the thesis. Secondly, we classify the point primitive non-trivial finite linear spaces with number of points being the product of two distinct primes. Next, based on the previous research, we extend the classification of line transitive point imprimitive linear spaces with the Fang-Li parameter gcd(k,r) up to10. Finally, the2-(106,6,1) design is been studied and we prove that there exists no point transitive2-(106,6,1) design.In Chapter3, based on the result of Liebeck and Saxl about the classification of primitive groups of order mp, where1≤m≤p and p is a prime, we classify the point primitive non-trivial finite linear spaces with number of points being the product of two distinct primes. Furthermore, given a small positive integer k≥3and a transitive group G on a set V, where|P|=v, we also present an algorithm to sift all2-(v, k,1) designs with point-set P, and admitting G as its automorphism group.Let S=(P, C) be a non-trivial linear space with nip points, where m and p are two primes with m<p. Assume that G≤Aut(S) is point primitive, and the common line size is k. We prove that S is the Desarguesian projective space PG(d-1,q)(d≥3), the Hermitian unital UH(q), or Soc(G)=PSL(2,p) with k<(1+(?)2p2-2p-3)/2and p≥11, acting on the cosets of dihedral subgroup H of G such that H∩Soc(G)=Dp+1with few exceptions. Moreover, if G is line transitive, then S is the Desarguesian projectivespace PG(d1, q)(d≥3) or the Hermitian unital UH(q).In Chapter4, by using the computer algebra systems MAGMA and GAP, we extendthe classification of line transitive point imprimitive linear spaces with the Fang-Li pa-rameter gcd(k, r) up to10on the basis of previous research. Let S be a non-trivial finitelinear space admitting a line transitive and point imprimitive automorphism group G, andthe Fang-Li parameter k(r)=gcd(k, r)=9or10. We prove that S is the Desarguesianprojective plane PG(2,9).Combining the result of Betten, Delandtsheer, Law, Niemeyer, Praeger and Zhoushenglin in2009, we obtain that if k(r)=gcd(k, r)≤10, then S is the Desarguesianprojective plane PG(2,4), PG(2,7) or PG(2,9), or the Mills design or Colbourn-McCalladesign, both with (v, k)=(91,6), or one of the467Nickel-Niemeyer-O’Keefe-Penttila-Praeger designs with (v, k)=(729,8).In Chapter5, we prove that the non-existence of point transitive2-(106,6,1) designsusing the general linear spaces theory.