Block-transitive automorphism groups of Steiner 3-designs (Q6172284)

A \(t\)-\((v,k,\lambda)\) design \(D\) is a pair \((X,B)\) where \(X\) is a \(v\)-set of points and \(B\) is a collection of \(k\)-subsets of \(X\) called blocks with the property that every \(t\)-subset of \( X\) is contained in precisely \(\lambda\) blocks. When \(\lambda = 1\), \(D\) is called a Steiner \(t\)-design. If \(t < k < v-t\), then \(D\) is called nontrivial. An automorphism of a \(t\)-design is a permutation of the points which preserves the blocks. An automorphism of a design is called block-transitive if it acts transitively on the blocks and point-primitive if it acts primitively on the points. The main result of this paper is to show that if \(G\) is a block-transitive point-primitive automorphism group of a nontrivial Steiner 3-design, then \(G\) is either affine or almost simple.