Let \({\mathcal {D}}\) be a nontrivial 3-(v, k, 1) design admitting a block-transitive group G of automorphisms. A recent work of Gan and the second author asserts that G is either affine or almost simple. In this paper, it is proved that if G is almost simple with socle an alternating group, then \({\mathcal {D}}\) is the unique 3-(10, 4, 1) design, and \(G=\textrm{PGL}(2,9)\), \(\textrm{M}_{10}\) or \(\textrm{Aut}(\textrm{A}_6 )=\textrm{S}_6:\textrm{Z}_2\), and G is flag-transitive.

中文翻译：

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与交替组关联的块传递 3-(v, k, 1) 设计

设\({\mathcal {D}}\)是一个非平凡的 3-( v ,  k , 1) 设计，它允许自同构的块传递群G。Gan 和第二作者最近的一项工作断言G要么是仿射的，要么几乎是简单的。在本文中，证明了如果G几乎是简单的并且有一个交替群，则\({\mathcal {D}}\)是唯一的 3-(10, 4, 1) 设计，并且\(G= \textrm{PGL}(2,9)\) , \(\textrm{M}_{10}\)或\(\textrm{Aut}(\textrm{A}_6 )=\textrm{S}_6: \textrm{Z}_2\)，并且G是标志传递的。