Block-transitive automorphism groups of 2-( v , k , λ ) $(v,k,\lambda )$ designs with ( r , k ) = 1 $(r,k)=1$

作者：Zhang, Wanbao; Zhou, Shenglin^*

来源：Journal of Combinatorial Designs, 2022, 30(5): 315-331.

DOI：10.1002/jcd.21826

• 2-design
• automorphism group
• block-transitivity
• primitivity

In 1984, Camina and Gagen gave the result that the block-transitive automorphism group G $G$ of a 2- ( v , k , 1 ) $(v,k,1)$ design with k divide v $k\,| \,v$ must be flag-transitive, moreover, G $G$ is point-primitive of affine or almost simple type. As a generalization of this result, the purpose of this paper is to study block-transitive automorphism groups of nontrivial 2- ( v , k , lambda ) $(v,k,\lambda )$ designs with ( r , k ) = 1 $(r,k)=1$ , where r $r$ is the number of blocks incident with a given point. We prove that, for a 2- ( v , k , lambda ) $(v,k,\lambda )$ design D ${\mathscr{D}}$ with ( r , k ) = 1 $(r,k)=1$ , if G <= A u t ( D ) $G\le Aut({\mathscr{D}})$ is block-transitive, then G $G$ must be flag-transitive, and furthermore, G $G$ is point-primitive of affine or almost simple type. Moreover, the classification of this type of 2-designs is given when the socle of G $G$ is sporadic. There are thirteen 2-designs up to isomorphism.