Each Invertible Sharply d-Transitive Finite Permutation Set with d ≥ 4 is a Group
Each Invertible Sharply d-Transitive Finite Permutation Set with d ≥ 4 is a Group
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每个 d ≥ 4 的可逆锐 d-传递有限排列集是一个群
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
P. Quattrocchi
中科院分区：
文献类型：
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作者：
A. Bonisoli;P. Quattrocchi
所有已知的包含单位元的有限锐4-传递置换集都是群，即S4，S5，A6和11次Mathieu群。证明了包含单位元的11元上的锐4-传递置换集必然是11次Mathieu群。证明使用直接计数参数。它基于11阶Mathieu群中对合的组合性质（在此建立）和9阶Minkowski平面的唯一性（之前已经建立）：这两个事实的有效性依赖于计算机计算。一个置换集被称为可逆的，如果它包含单位元，并且如果每当它包含一个置换时，它也包含它的逆。在由可逆置换集产生的几何结构中，至少有一个块对称是自同构。上述结果具有以下后果。i）包含单位元的12个元素上的5-可迁置换集必然是12度的马蒂厄群。ii）不存在13个元素上的6-可迁置换集。当d ≥ 6时，至少有d + 3个元素的有限集上不存在可逆的锐d-传递置换集。iii）一个有限可逆的d-可迁置换集，当d ≥ 4时，必然是一个群，它要么是对称群，要么是交错群，要么是11次Mathieu群，要么是12次Mathieu群。
All known finite sharply 4-transitive permutation sets containing the identity are groups, namely S4, S5, A6 and the Mathieu group of degree 11. We prove that a sharply 4-transitive permutation set on 11 elements containing the identity must necessarily be the Mathieu group of degree 11. The proof uses direct counting arguments. It is based on a combinatorial property of the involutions in the Mathieu group of degree 11 (which is established here) and on the uniqueness of the Minkowski planes of order 9 (which had been established before): the validity of both facts relies on computer calculations. A permutation set is said to be invertible if it contains the identity and if whenever it contains a permutation it also contains its inverse. In the geometric structure arising from an invertible permutation set at least one block-symmetry is an automorphism. The above result has the following consequences. i) A sharply 5-transitive permutation set on 12 elements containing the identity is necessarily the Mathieu group of degree 12. ii) There exists no sharply 6-transitive permutation set on 13 elements. For d ≥ 6 there exists no invertible sharply d-transitive permutation set on a finite set with at least d + 3 elements. iii) A finite invertible sharply d-transitive permutation set with d ≥ 4 is necessarily a group, that is either a symmetric group, an alternating group, the Mathieu group of degree 11 or the Mathieu group of degree 12.