Flag-transitive, point-imprimitive symmetric \(2\)-\((v, k, \lambda)\) designs with \(k > \lambda (\lambda - 3) / 2\). (English) Zbl 1541.05023
Summary: Let \(\mathcal{D} = (\mathcal{P}, \mathcal{B})\) be a symmetric 2-\((v, k, \lambda)\) design admitting a flag-transitive, point-imprimitive automorphism group \(G\) that leaves invariant a non-trivial partition \(\Sigma\) of \(\mathcal{P}\). C. E. Praeger and S. Zhou, J. Comb. Theory, Ser. A 113, No. 7, 1381–1395 (2006; Zbl 1106.05012)] have shown that, there is a constant \(k_0\) such that, for each \(B \in \mathcal{B}\) and \(\Delta \in \Sigma\), the size of \(|B \cap \Delta |\) is either 0 or \(k_0\). In the present paper we show that, if \(k > \lambda (\lambda - 3) / 2\) and \(k_0 \geqslant 3\), \(\mathcal{D}\) is isomorphic to one of the known flag-transitive, point-imprimitive symmetric 2-designs with parameters \((45, 12, 3)\) or \((96, 20, 4)\).
MSC:
| 05B30 | Other designs, configurations |
| 05B05 | Combinatorial aspects of block designs |
| 20D45 | Automorphisms of abstract finite groups |
| 20B10 | Characterization theorems for permutation groups |
| 20B30 | Symmetric groups |
Citations:
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