Chain-imprimitive, flag-transitive 2-designs (Q6591957)

In [\textit{C. Amarra} et al., J. Comb. Theory, Ser. A 205, Article ID 105866, 32 p. (2024; Zbl 1536.05042)] it was shown that there exist \(2\)-designs with a block-transitive automorphism group \(G\) such that the action on points is \(s\)-chain-imprimitive for arbitrarily large chain length \(s\). In the present paper, it is proved that the same assertion holds true if \(G\) is flag-transitive, not only block-transitive. Necessary and sufficient conditions on the parameters of a \(2\)-design with such an automorphism group are obtained in Theorem 1.2. The proof of this theorem relies on an explicit construction, and for the main result, the authors display infinite series of parameters satisfying the conditions for each \(s\ge 2\). For \(s=3\), \(v=e_1 e_2 e_3\), and \(e_1,e_2,e_3\le 50\) many more parameters satisfying the conditions are found by an exhaustive computer search in MAGMA. This shows that there could be other infinite families of flag-transitive \(s\)-chain imprimitive \(2\)-designs.