Double Pass Group With 4 - (v, K, 2) Design,

This paper mainly study 2-transitive permutation groups and the non-trivial designs by the classifical methods. We want to find out the flag-transitive designs. From 2001 to 2005, Michael Huber had characterized all flag-transitive Steiner-3 designs and Steiner-4 designs, mainly by the use of the O'Nan-Scott theorem,the classification of finite simple groups,the classification of the finite doubly transitive permutation groups and some relevant design theories. This paper mainly base on the work of Michael Huber and further conside the non-trivial 4-(v, k,2) designs, and obtained a series of new results.Main Theorem 1:Let D= (X,B) be a non-trivial 4-(v,k,2) design, G≤Aut(D). If G is a flag-transitive affine type group, then G (or Go) and v is neither of:(ⅰ) G0(?)>SL(-,pa), d≥2a, v= pd;(ⅱ) G≤A(?)L(1,v), v= pd;(ⅲ) G0(?)G2(2a)', d= 6a, v= 2d.Main Theorem 2:Let D= (X,B) be a non-trivial 4-(v,k,2) design, G≤Aut(D). If G is a flag-transitive almost simple type group, then N and v is neither of:(A)Av,v≥5;(B) Sz(q),v=q2+1,q=2e+1>2; (C)Re(q),v=q3+1,q=32e+1>3；(D)PSL(2,11),v=11,k≠6；(E)PSL(2,8),v=28；(F)A7,V=15；(G)Mv,v=12,22,23,24.