Block-Transitive \(2\)-\((v, k, 1)\) Designs and the Groups \(E_7(q)\)

Abstract

This article is a contribution to the study of block-transitive automorphism groups of \(2\)-\((v,k,1)\) block designs. Let \(\mathcal{D}\) be a \(2\)-\((v,k,1)\) design admitting a block-transitive, point-primitive but not flag-transitive group \(G\) of automorphisms. Let \(k_1 = (k, v-1)\) and \(q = p^f\) for prime \(p\). In this paper we prove that if \(G\) and \(D\) are as above and \(q > {(2(k_rk-k_r+1)f)^{\frac{1}{4}}}\) then \(G\) does not admit a Chevalley group \(E_7(q)\) as its socle.