Thirty years ago, a six-person team classified the pairs   (D,G)      (  D  ,  G  )         where   D     D        is a   2−(v,k,1)      2  −  (  v  ,  k  ,  1  )         design and  G    is a flag-transitive automorphism group of   D,      D  ,         with the exception of those in which  G    is a one-dimensional affine group. Since then the effort has been to classify those   D,      D  ,         designs which are block-transitive but not flag-transitive. This paper contributes to the program for determining the pairs   2−(v,k,1)      2  −  (  v  ,  k  ,  1  )         in which   (D,G)      (  D  ,  G  )         has a block-transitive group  G    of automorphisms. It is clear that if one wishes to study the structure of a finite group acting on a   D     D        design then describing the socle is an important first step. Here we prove that if  G    is a block-transitive group of automorphisms of   2−(v,k,1)      2  −  (  v  ,  k  ,  1  )         which has   D     D        as its socle then  T    is also transitive on the blocks of   T= 2 F 4 (q)      T  =      2    F  4    (  q  )          .
['Shangzhao Li', 'Weijun Liu', 'Xianhua Li']
On the automorphisms of 2−(v,k,1) designs