Flag-transitive non-symmetric 2-designs with \((r,\lambda)=1\) and alternating socle (Q2341052)

Summary: This paper deals with flag-transitive non-symmetric 2-designs with \((r,\lambda)=1\). We prove that if \(\mathcal D\) is a non-trivial non-symmetric \(2-(v,k,\lambda)\) design with \((r,\lambda)=1\) and \(G\leq \operatorname{Aut}(\mathcal D)\) is flag-transitive with \(\operatorname{Soc}(G)=A_n\) for \(n\geq 5\), then \(\mathcal D\) is a \(2-(6,3,2)\) design, the projective space \(\operatorname{PG}(3,2)\), or a \(2-(10,6,5)\) design.