Flag-transitive non-symmetric 2-designs with \((r,\lambda)=1\) and sporadic socle. (English) Zbl 1347.05019
Summary: This paper is a contribution to the investigation of flag-transitive non-symmetric 2-designs. We prove that if \(\mathcal {D}\) is a non-trivial non-symmetric 2-\((v,k,\lambda)\) design with \((r,\lambda)=1\) and \(G\leq \mathrm{Aut}(\mathcal {D})\) is flag-transitive with sporadic socle, then \(\mathcal {D}\) must be one of following: a 2-(12, 6, 5) design with \(G=M_{11}\), or a 2-(22, 6, 5) design with \(G=M_{22}\) or \(M_{22}:2\).
MSC:
| 05B05 | Combinatorial aspects of block designs |
| 05B25 | Combinatorial aspects of finite geometries |
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
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