| Block-Transitive $2$-$(v,k,1)$ Designs and Groups $E_6(q)$ |
| Received:November 11, 2008 Revised:May 15, 2009 |
| Key Words:
block design block-transitive point-primitive automorphism group.
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10871205), China Postdoctoral Science Foundation Funded Project (Grant No.20080441323) and Scientific Research Fund of Zhejiang Education Department (Grant No.Y200804780). |
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| Abstract: |
| This article is a contribution to the study of block-transitive automorphism groups of $2$-$(v, k, 1)$ block designs. Let ${\cal D}$ be a $2$-$(v, k, 1)$ design admitting a block-transitive, point-primitive but not flag-transitive automorphism group $G$. Let $k_r=(k,v-1)$ and $q=p^f$ for prime $p$. In this paper we prove that if $G$ and ${\cal D}$ are as above and $q>$ $(3(k_rk-k_r 1)f)^{1/3}$, then $G$ does not admit a simple group $E_6(q)$ as its socle. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2010.04.002 |
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