Symmetric designs and finite simple exceptional groups of Lie type
Seyed Hassan Alavi, Mohsen Bayat, Ashraf Daneshkhah · arXiv (Cornell University) · 2017
In this article, we study symmetric $(v, k, \lambda)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is a finite simple exceptional group of Lie type. We prove a reduction theorem to some possible parameters of such designs. In particular, if $\lambda\leq 100$, we show that there are only two such designs, namely, $(351,126,45)$ and $(378,117,36)$ for $G=G_{2}(3)$. We also find symmetric designs with parameters $(36,21,12)$ and $(63,32,16)$ and flag-transitive and point-primitive automorphism group $G=G_{2}(2)$ of rank three and four, respectively. As a main tool to this investigation, part of this paper is devoted to studying subdegrees and large maximal subgroups of almost simple groups whose socle is a finite simple exceptional group of Lie type.