\(2\)-\((v,k,1)\) designs and PSL\((3,q)\) where \(q\) is odd (Q1430967)

Let \(V\) be a finite set of \(v\) elements and \(B\) be a collection of some subsets of \(V\), the subsets are called blocks. A \(2\)-\(( v, k, 1)\) design \(D= (V,B)\) is that configuration in which each block in \(B\) contains exactly \(k\) elements and any two elements of \(V\) appear exactly in one block. An automorphism of \(D\) is a permutation of elements which leaves the set \(B\) invariant, the group formed by all the automorphisms of \(D\) is denoted by \(\Aut D\). A subgroup \(G\) (of \(\Aut D\)) is called point-transitive (point-primitive) if \(G\) is transitive (primitive) on \(V\), and it is said to be block-transitive if \(G\) is transitive on \(B\). The main result of this paper is the following result: Let \(G\) be a block-primitive automorphism group of a \(2\)-\((v, k, 1)\) design. If \(G\) is isomorphic to \(\text{PSL}(3,q)\), \(q\) being odd, then \(G\) is also point-primitive.