On groups with several doubly-transitive permutation representations
On groups with several doubly-transitive permutation representations
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关于具有多个双传递排列表示的群
DOI：
10.1007/bf01111509
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发表时间：
1972
影响因子：
0.8
通讯作者：
P. Cameron
中科院分区：
文献类型：
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作者：
P. Cameron
具有两个不等价（有限）双传递置换表示且具有相同置换特征标的群是射影设计的自同构群。因此，一个具有两个以上这样的表示的群是通过一次取两个表示而得到的每个设计的自同构群，并且保持这些设计之间的相互关系。从这种情况下抽象的组合对象，暂时称为一个系统的联系投影设计，进行了研究，并将结果应用到组。在第一节中，描述了投射设计的一些基本性质，以及由群的两个双传递表示构造投射设计。第2节定义了关联投射设计的系统，并开始对这些系统进行组合分析，建立连接它们的参数的关系。在第三节中，我们注意到这些关系的一个简单的推论，并由此导出了双传递群的自同构的一些结果。(This第一节的灵感来自Wielandt的论文I-8]关于同一主题。第4节给出了一组由三个链接设计组成的系统的参数所满足的方程。我得出这些方程没有后果，但我给他们希望，他们可能是有用的，在进一步的工作就这个问题。第5节包含一个帐户的应用到一个猜想的Wielandt（素数度p的传递置换群至多有两个度p的不等价表示）;虽然我不能证明这个猜想，但这些方程很适合计算，彼得·诺依曼博士用它们证明了这个猜想对小于200万的素数是正确的。我很感谢诺依曼博士的大量研究。的时间和精力，他已经投入到这一计算，并为许多有益的讨论;和教授Wielandt，他的兴趣，这项工作，为许多有益的建议，提高了清晰度本文，并为几个未发表的结果。我还要感谢我的妻子（正是在我们度蜜月的时候，我想到了这个问题）;感谢牛津大学默顿学院，我在那里获得了初级研究奖学金。
A group which has two inequivalent (finite) doubly transitive permutation representations with the same permutation character is an automorphism group of a projective design. So a group with more than two such representations is an automorphism group of each of the designs obtained by taking the representations two at a time, and preserves interrelations among these designs. The combinatorial object abstracted from this situation, tentatively called a system of linked projective designs, is studied, and the results applied to the groups. In section 1, some basic properties of projective designs, and their construction from two doubly transitive representations of a group, are described. Section 2 defines systems of linked projective designs, and begins a combinatorial analysis of these systems, establishing relations connecting their parameters. In section 3, a simple consequence of these relations is noted, and some results on automorphisms of doubly transitive groups are deduced from it.(This section was inspired by Wielandt's paper I-8] on the same topic.) Section 4 gives a collection of equations satisfied by the parameters of a system of three linked designs. I derive no consequences of these equations, but I give them in the hope that they may be useful in further work on this subject. Section 5 contains an account of an application to a conjecture of Wielandt (that a transitive permutation group of prime degree p has at most two inequivalent representations of degree p); while I cannot prove the conjecture, the equations are well suited to computation, and Dr. Peter Neumann has used them to show that the conjecture is true for primes less than two million.I am grateful to Dr. Neumann for the large amount of time and energy he has devoted to this computation, and for many helpful discussions; and to Professor Wielandt, for his interest in this work, for many helpful suggestions which improved the clarity of this paper, and for several unpublished results. Also I acknowledge a debt to my wife (it was on our honeymoon that this line of attack on the problem occurred to me); and to Merton College, Oxford, where I hold a Junior Research Fellowship.