Almost simple groups of Lie type and symmetric designs with \(\lambda\) prime. (English) Zbl 1464.05020
Summary: In this article, we investigate symmetric \((v,k,\lambda)\) designs \(\mathcal{D}\) with \(\lambda\) prime admitting flag-transitive and point-primitive automorphism groups \(G\). We prove that if \(G\) is an almost simple group with socle a finite simple group of Lie type, then \(\mathcal{D}\) is either the point-hyperplane design of a projective space \(\mathrm{PG}_{n-1}(q)\), or it is of parameters \((7,4,2)\), \((11,5,2)\), \((11,6,3)\) or \((45,12,3)\).
MSC:
| 05B05 | Combinatorial aspects of block designs |
| 05B25 | Combinatorial aspects of finite geometries |
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
| 05E14 | Combinatorial aspects of algebraic geometry |
Software:
GAPReferences:
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