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In 1988, Delandtsheer and Doyen conjectured that for a 2-(v,k, 1) design D, if Gis a block-primitive automorphism group of D then G is also point-primitive. There aremany results about this conjecture. In 1989, Delandtsheer proved that if G is block-primitive then G is an afine or almost simple group. In 1993, Weidong Fang and HuilingLi introduced a set of parameters and they proved Delandtsheer-Doyen conjecture holdwhen k(r) = (k,r)≤4. In 1996, Weijun Liu proved that if k(r) = (k,r)≤10 Delandtsheer-Doyen conjecture hold. In 2006, Yanbo Ma extended the bound up to k(r) = (k,r)≤12.With the methods in these papers, we introduce some new methods with which wenot only prove Delandtsheer-Doyen conjecture hold when k(r) = (k,r)≤14, but alsoprovide new methods for solving the problems on block-primitive 2-(v,k, 1) designs.Our main results is as following:Main Theorem: Assume that D is a 2-(v,k, 1) design, if G≤Aut(D) is block-primitive and (k,r)≤14, then G is also point-primitive.The following is the structure of this thesis:We introduce the research background in chapter I. first, we give some basic defi-nitions on 2-(v,k, 1) designs, introduce the development history and research status ofcombinatorial of designs and theirs automorphism groups, then we talk about the originof the Delandtsheer-Doyen conjecture, and explain the significance of this conjecture.In Chapter II, we obtain 203 9-tuples under the hypothesis that the automorphismgroup 9-tuple of D is block-primitive, not point-primitive. Then we discuss these 9-tuplescase by case to eliminate 202 9-tuples except one 9-tuple.In the following chapter, we rule out the 100th 9-tuple. Because in this case theparameters are large while its prime factors are small, so we can’t rule out this 9-tuple bythe methods used in Chapter II. Here we use the classification of the primitive groups ofodd degree to rule out this 9-tuple with tedious computation, finally we prove our MainTheorem.
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