Block-transitive Steiner t-designs and sporadic simple groups(区传递Steiner t-设计与散在单群)

Abstract

The study of t-designs with certain highly symmetric properties and the classification of their automorphism groups is one of the important topics of algebraic combinatorics. Hitherto the classification of flag-transitive Steiner t-designs has been completely settled by Huber. Nevertheless, it is interesting that the results of t-designs will get richer when G-flag-transitivity are weakened to G-block-transitivity and the research work becomes more challenging as well. Depending on the classification of finite simple groups and 2-transitive permutation groups, we show the classification of block-transitive Steiner t-designs admitting a block-transitive automorphism group with sporadic socle, and draw a conclusion as follows: Let D=(P,E) be a non-trivial Steiner t-design and G≤Aut(D) be point-primitive. If the socle of G is a sporadic simple group, then G acts block-transitively on D if and only if G acts flag-transitively on D, and D is one of the five known flag-transitive Steiner t-designs.(研究具有某种高度对称性的t-设计及其自同构群的分类问题是代数组合学的重要课题之一。目前关于旗传递Steiner t-设计的分类问题已被Huber解决。然而当设计的自同构群的旗传递性减弱为区传递时，研究结果更加丰富，工作也更加困难。利用有限单群分类定理及2-传递置换群的分类结果，解决了自同构群为散在单群的区传递Steiner t-设计的分类问题，并证明了：设D=(P,E)为非平凡Steiner t-设计且G≤Aut(D)为点本原群，若G的基柱Soc(G)为散在单群，则G区传递作用在D上当且仅当G旗传递作用在D上，且D为已知的5种旗传递Steiner t-设计之一。)

Keywords

• steiner t-designs(steiner t-设计)
• block-transitive(区传递)
• point-primitive(点本原)
• sporadic simple group(散在单群)