Mathieu群与旗传递2-(v,k,λ)设计

Mathieu Groups and Flag-Transitive 2-(v,k,λ) Designs

旗传递性是群作用在2-(v,k,λ) 设计上的重要性质之一。对满足一定条件的旗传递2-设计进行分类是一个比较有意思的问题。Dembowski已经证明了满足条件(v-1,k-1)≤2 的旗传递2-(v,k,λ) 设计的自同构群G是本原群。据此,本文在条件(v-1,k-1)≤2 下,研究自同构群旗传递且其基柱Soc(G)是五个Mathieu 群之一时的2-(v,k,λ) 设计的分类问题,得到了在同构意义下存在62个这样的设计。

Flag-transitivity is one of the important conditions that can be imposed on the automorphism group of a 2-(v,k,λ) design. The classification of flag-transitive 2-designs is an important problem in the algebraic combinatorial theory. Dembowski has proved that if G≤Aut(D) is flag-transitive and (v-1,k-1)≤2, then G is also point-primitive. According to this result, in this paper we completed the classification of this type of designs, with Soc(G) was one of five Mathieu groups Mi, where i=11, 12, 22, 23 or 24. We prove that there exists 62 2-designs satisfying the assumption.