Block-transitive \(3\)-\((v, k, 1)\) designs associated with alternating groups (Q6109714)

A \(t\)-\((v,k,\lambda)\) design is an incidence structure consisting of a set \(\mathcal P\) of \(v\) points and a set \(\mathcal B\) of \(k\)-subsets of \(\mathcal P\), called blocks, such that each block in \(\mathcal B\) has size \(k\), and each \(t\)-subset of \(\mathcal P\) lies in exactly \(\lambda \) blocks from \(\mathcal B\). A design \(\mathcal D\) is said to be trivial if \(\mathcal B\) consists of all the \(k\)-subsets of \(\mathcal P\). A flag of \(\mathcal D\) is a pair \((\alpha,B)\), where \(\alpha\) is a point and \(B\) is a block containing \(\alpha\). An automorphism of \(\mathcal D\) is a permutation on \(\mathcal P\) which permutes the blocks among themselves. For a subgroup \(G\) of the automorphism group Aut\((\mathcal D)\) of \(\mathcal D\), the design \(\mathcal D\) is said to be \(G\)-block-transitive if \(G\) acts transitively on the set of blocks and is said to be block transitive if it is Aut\((\mathcal D)\)-block-transitive. The point-transitivity and flag-transitivity are defined in a similar way. There has been an extensive study on the classification of point-transitive, flag-transitive, and block-transitive designs, mainly in the case of \(2\)-\((v,k,1)\) designs. There are fewer results for \(t\)-\((v,k,1)\) designs for \(t\in \{3,4,5\}\). In this context, it has been proved recently [\textit{Y. Gan} and \textit{W. Liu}, Discrete Math. 346, No. 10, Article ID 113534, 7 p. (2023; Zbl 1521.05012)] that, for a nontrivial \(G\)-block-transitive \(3\)-\((v,k,1)\) design, the group \(G\) is either affine or almost simple. So the authors suggest the following problem: Problem. Classify nontrivial \(G\)-block-transitive \(3\)-\((v,k,1)\) designs, where \(G\) is an almost simple group. In this paper, the authors focus on the case that \(G\) is an almost simple group with alternating socle \(A_n\). The main result of this paper is the following: Theorem. Let \(G\) be an almost simple group with alternating socle \(A_n\), \(n\ge 5\). Suppose that \(D\) is a nontrivial \(G\)-block-transitive \(3\)-\((v,k,1)\) design. Then \(G=\operatorname{PGL}_2(9)\), \(M_{10}\) or \(S_6\colon Z_2\), and \(\mathcal D\) is \(G\)-flag-transitive with parameters \(v=10\) and \(k=4\).