Download the PDF of the paper entitled Block Transition $3$-$(v,k,1)$ Designs for Lie-type Exceptional Groups by Ting Lan, Weijun Liu, and Fu-Gang ying.

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overview: Let $\mathcal{D}$ be a nontrivial $G$-block-transitive $3$-$(v,k,1)$ design. where $T\leq G \leq \mathrm{Aut}(T) $ is for some finite non-Abelian simple group $T$. If $T$ is a simple exception group of Lie type, then $T$ is proved to be either the Suzuki group ${}^2B_2(q)$ or $G_2(q)$. Moreover, for $T={}^2B_2(q)$ , the plan $\mathcal{D}$ contains the parameters $v=q^2+1$ and $k=q+1$ , so $ \mathcal{ D}$ is the inverse plane of order $q$. If $T=G_2(q)$, the point stabilizer for $T$ is either $\mathrm{SL}_3(q).2$ or $\mathrm{SU}_3(q).2$. Become. The parameter $k$ satisfies very restrictive conditions.

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