Groups And T-designs With Large T

Three main fields of work have been done by us based on the conclusions drawn in this paper.First, we construct3-designs using the orbits number and its length of the action of PSL(2,q)(q≡1(mod4)) on the projective line X=GF(q)∪{∞}, which therefore determines the stabilizer of the k-subsets of X.During the discussion of this problem, we use the orbits number and its length of the action of PSL(2,q)(q≡1(mod4)) on X|k|, considering the stabilizer GL of the block L, here GL≡{g∈G|Lg=L,L∈B}. With the the orbits number and its length of L, we use the method of classification which is the basic knowledge, determining the stabilizer GL of the block L, and constructs some new simple3-designs (the main theorem1).1(1) Let q≡1or13(mod16) and q>22. Let k=q-1/4, then there exists an non-trivial3-(q+1,k,(k-1)(k-2)/2) design with PSL(2,q).(2) Let q≡1or5(mod16), q≥16. let k=(q-1)/4+2, then there exists an non-trivial3-(q+1,k, k(k-1)/2design with PSL(2, q).(3) Let q=28r+1be an odd power of a prime p, then there exists a simple3-(q+1,7,15) design with PSL(2,q).(4)Let q=44r+1be an odd power of p, then there exists a 3-(q+1,11,45) design with PSL(2,q)(5)Let q=24r+lbe an odd power of p and p≠13, then there exists a3-(q+1,13,143) design with PSL(2,q).Second, we discuss the existence of4-(v,5,λ) designs with a almost simple group with PSL(2,q) as its socle.We concern on the existence of this type of designs. Here, we mainly use some number of the definition of designs and it determine the possible number of p and q, where q=pe is a prime power, then we get:(1)q=17, then we have G=PSL(2,17) or PGL(2,17).Using the basic number theorem and designs and the computer, we construct designs and prove that there only exists4-(18,5,4) designs.(2) g=32, we have G=PSL(2,32)=PGL(2,32) or PTL(2,32).The method is similar to (1), we get that there only exists4-(33,5,4),4-(33,5,20)å'Œ4-(33,5,5) design, that is, proving the main theorem2:let D=(X,B) be a4-(q+1,5,λ) design, G≤Aut(D) acts block-transitive on D and X=GF(q)∪{∞}, GF(q)is a finite field of order q. let PSL(2,q)(?)G≤PTL(2,q), then there only exists4-(33,5,4),4-(18,5,4),4-(33,5,20),4-(33,5,5) designs.Third, if G is the flag-transitive automorphism group of a non-trivial5-(v,k,2) design D, then Soc(G)=PSL(2,q), here q=2n or3". The condition of had been discussed. Being the continuation of this part of work, we talk about the condition of q=3n.We talk with the two conditions of G=G*and|G:G*|=2, where G*=G∩(PSL(2,q)):(τα), That is, G*=PSL{2,q):(G∩<Tα>).During the discuss, we use the theorem of groups, designs and some known result and get the main theorem3: let D=(X, B) be a non-trivial5-(v,k,2) designs, if Soc (G)=PSL (2,2n), then G can not act flag-transitive on D.