Solvable block-transitive automorphism groups of \(2\)-\((v,k,1)\) designs (Q5929843)

The classification of \(2\)-\((v,k,1)\) designs with flag-transitive automorphism groups has been completed [see \textit{F. Buekenhout} et al., Geom. Dedicata 36, No. 1, 89-94 (1990; Zbl 0707.51017)]. This paper can be seen as a continuation of the work in [Geom. Dedicata 36, 89-94 (1990)] and [\textit{A. R. Camina}, Bull. Lond. Math. Soc. 28, No. 3, 269-272 (1996; Zbl 0855.20002)]. This paper considers the designs with block-transitive automorphisms groups, and provides the following theorem: ``Let \(k\geq 3\) be a fixed positive integer and \((G,{\mathcal D})\) be a pair, where \({\mathcal D}\) is a \(2\)-\((v,k,1)\) design and \(G\) is a group of automorphisms of \({\mathcal D}\) such that \(G\) is solvable and block-transitive on \({\mathcal D}\). If \(v> (k^{3/4}+ 1)^{\phi(k(k- 1))}\), then \(v\) is a power of a prime number \(p\) and \(G\) is flag-transitive or \(G\leq \text{A}\Gamma\text{L}(1,v)\).'' (\(\phi(n)\) is the value of the Euler function at \(n\).) The proof of the above theorem is based on the Hering theory of solvable linear groups.