AI 中文总结
本文引入端点k路径传递图，在测地线情形下通过Hasse图归约为真块，并分类仿射与几乎单情形下的真块，给出稳定子分解准则以保持端点测地线传递性。
AI 中文摘要
我们引入端点k路径传递图，其中固定两个指定顶点的自同构群逐点作用在连接它们的长度为k的路径上传递。我们研究测地线情形，即k等于这两个顶点之间的距离。我们将这些测地线的并集与有限有界分次偏序集的Hasse图等同，并通过完全割和匹配压缩将其归约为真块。假设诱导群K在第一距离层上忠实且2-齐次作用。我们首先证明，K的每个非平凡正规子群在每个内部层上传递当且仅当soc(K)如此。在此正规基本条件下，我们分类仿射情形下的真块以及几乎单情形下具有2传递内部层的等宽真块。非平凡设计界面在仿射情形下为Paley或仿射辛设计，在几乎单情形下为射影设计、2-(11,5,2)设计、Higman-Sims设计或其补设计。真块具有至多四的约化秩，而几乎单真块在无等宽条件下可具有任意大的约化秩。我们还给出一个稳定子分解准则，用于在保持端点测地线传递性的同时组装块。
英文摘要
We introduce endpoint k-path-transitive graphs, in which the automorphism group fixing two prescribed vertices pointwise acts transitively on the paths of length k joining them. We study the geodesic case, where k equals the distance between these vertices. We identify the union of these geodesics with the Hasse graph of a finite bounded graded poset and reduce it to proper blocks by complete cuts andmatching compression. Assume that the induced group K acts faithfully and 2-homogeneously on the first distance layer. We first prove that every nontrivial normal subgroup of K is transitive on every internal layer if and only if soc(K) is. Under this normal-basic condition, we classify the proper blocks in the affine case and the equal-width proper blocks with a 2-transitive internal layer in the almost-simple case. The nontrivial design interfaces are Paley or affine symplectic designs in the affine case, and projective designs, the 2-(11, 5, 2) design, the HigmanSims design, or their complements in the almost-simple case. The proper blocks have reduced rank at most four, whereas almost-simple proper blocks can have arbitrarily large reduced rank without the equal-width condition. We also give a stabilizer factorization criterion for assembling blocks while preserving endpoint-geodesic transitivity.
