区传递的$2-(v, k, 1)$ 设计与单群$E_6(q)$

Block-Transitive $2$-$(v,k,1)$ Designs and Groups $E_6(q)$

中文关键词:  设计  区传递  点本原  自同构群.

英文关键词:block design  block-transitive  point-primitive  automorphism group.

中文摘要:

      讨论区传递的$2-(v, k, 1)$设计的分类问题.特别地, 讨论自同构群的基柱为李型单群$E_6(q)$的区传递$2-(v, k, 1)$设计,得到如下结论: 设${\cal D}$为 一个$2-(v, k, 1)$设计, $G\leq$Aut$(\cal D)$是区传递, 点本原但非旗传递的. 若 $q>$ $(3(k_rk-k_r 1)f)^{1/3}$(这里$k_r=(k,v-1)$, $q=p^f$, $p$是素数, $f$是正整数), 则 ${\rm Soc}(G)\not\congE_6(q)$.

英文摘要:

      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.