Flag-transitive automorphism groups of \(2\)-designs with \(\lambda \geq (r, \lambda )^2\) are not product type (Q6612119)

A \textit{\(2\) -- \((v, k, \lambda)\) design} \(\mathcal{D}\) is a pair \((\Omega, \mathcal{B})\) such that \(\Omega\) is a set of \(v\) points and \(\mathcal{B}\) is a set of blocks, that is, each block is a \(k\)-subset of \(\Omega\) and any two distinct points are contained in precisely \(\lambda\) blocks. The number of blocks containing a given point does not depend on the point, and we denote this integer by \(r\). The \textit{automorphism group of \(\mathcal{D}\)} consists of all the permutations of \(\Omega\) that preserves set-wise \(\mathcal{B}\). A subgroup \(G\) of the automorphism group of \(\mathcal{D}\) is \textit{flag-transitive} if it is transitive on the pairs \((\alpha,B)\in \Omega \times \mathcal{B}\), where \(\alpha\) and \(B\) are incident.\N\NIn [Ars Math. Contemp. 14, No. 1, 187--195 (2018; Zbl 1400.05035)], \textit{S. Zhou} and \textit{X. Zhan} proved that, if \(G\) is a flag-transitive group of automorphisms of \(\mathcal{D}\) with \(\lambda \ge (r, \lambda)^2\), then \(G\) is a primitive permutation group on \(\Omega\), and \(G\) is either of affine type, or of almost simple type, or of product action type. The present paper shows that product action type can never occur. These results refine a classical result of \textit{P. Dembowski} [Finite geometries. Berlin-Heidelberg-New York: Springer-Verlagn (1968; Zbl 0159.50001)].\N\NThe novel result of this paper has a clear strategy proof. Subsection ~2.1 has an arithmetic flavour, deriving bound on the parameters of the \(2\)-design \(\mathcal{D}\). In Subsection~2.2, an analysis of the structure of a block-stabilizer yields a lower bound on the size of the automorphism group of the socle of \(G\). Since the two direct factor \(T\) of the socle of \(G\) are isomorphic nonabelian \(2\)-transitive permutation groups, the automorphism groups of the socle of \(G\) is known, and its order is too small. This final contradiction is explained in Subsection~2.3.\N\NFinally, in Section~3, an infinite family of \(2\)-designs whose automorphism groups are of product action type is built.