Block-transitive Designs And The Sporadic Simple Group M₁₂

Since 1981,after the classification of the finite simple groups was completed,many scholars begin to study the combinatorial design by using the properties of the finite simple groups.The classification of designs is a leading subject of the finite group theory and combinatorial design theory.In recent years,many schoolers’s attentions have turned to the classification of block-transitive designs with the automorphism group of almost simple.In 2000,Camina and Spiezia considered the designs with the 26 sporadic simple groups as the socle of the automorphism group G and proved that there exist no such block-transitive 2-(v,k,1)design.In 2007,Han Guangguo gave the classification of block-transitive 2-(v,k,1)designs with the simple groups E8(q)of Lie type as the socle of the automorphism group.In 2013,Han Guangguo and Li Chuangui discussed the classifica-tion of point-primitive block-transitive but not flag-transitive 2-(v,11,1)designs with the classic simple group as the socle of the automorphism group.In 2016,Liang Hongxue and zhou Shenglin gave the classification of flag-transitive point-primitive 2-(v,k,2)de-signs with the simple groups as the socle of automorphism group.In 2016,Wang Beijun,Liang Hongxue and Zhou Shenglin gave the classification of flag-transitive point-primitive non-symmertic 2-(v,k,3)designs with the alternating socle An(n ≥5)as the socle of au-tomorphism group.In 2017,Zhou Shenglin and Zhang Xiaohong gave the classification of point-primitive block-transitive designs with the simple groups as the socle of the auto-morphism group.This paper study the classification of block-transitive point-primitive of 2-(v,k,λ)designs with simple groups as the socle of the automorphism group for λ ≥ 3.In this paper,we choose one of the 26 sporadic simple groups to study and consider the classification of block-transitive designs with the sporadic simple group M12 as the socle of the automorphism group.The main structure of this paper is as the following aspects:At first,we briefly intro-duce research background,research status and the research contents of this thesis.Next,We list some concepts of group theory,some concepts of combinatorial theory,and the the-orem of the group and the combinatorial theory.It provides a strong basis for the proof of the following theorem.At last,with the methods of Zhang Xiaohong and Zhou Shenglin’s classification of point-primitive 2-(v,k,2)designs with the sporadic simple groups as the socle of the automorphism group,we discuss the classification of block-transitive designs with the sporadic simple group M12 as the socle of the automorphism group.Then,we get the results:Main Theorem Let D be a nontrivial 2-v,k,λ)design,G≤<Aut(D)act block-transitively and point-primitively.We consider sporadic socle M12 of G and prove that if 3 ≤λ≤ 31,then(i)D must be 2-(144,13,12)design or 2-(144,66,30)design,G = M12 and Gα= L2(11);(ii)D must be a 2-(144,66,30)design,G = M12:2 and Ga =L2(11):2.