几乎单群与区传递Steiner 5-设计

Almost Simple Groups and Block-Transitive Steiner 5-Designs

摘要: 研究具有几乎单型的区传递自同构群的区传递Steine 5-设计的分类问题。利用二齐次置换群分类定理证明了：一个具有几乎单型区传递自同构群$G$ 的Steiner 5-设计$D$ ，则要么是一个$5\text{-}\left(12,6,1\right)$ 设计且自同构群$G\cong M_{12}$ 或者是一个$5\text{-}\left(24,8,1\right)$ 且自同构群$G\cong M_{24}$ ，要么设计的自同构群$G$ 的基柱$\text{Socle}\left(G\right)$ 只能是典型单群。

Abstract: The classification problem of block-transitive Steiner 5-designs with an almost simple block-transitive automorphism group is studied. By using the classification theorem of 2-homogeneous permutation groups, it is proved that if the Steiner 5-design has an almost simple block-transitive automorphism group, then either it is a $5\text{-}\left(12,6,1\right)$ design and the automorphism $G\cong M_{12}$ or it is a $5\text{-}\left(24,8,1\right)$ design and the automorphism $G\cong M_{24}$ , or the socle of the automorphism $G$ , $\text{Socle}\left(G\right)$ can only be a classical simple group.