On graph-restrictive permutation groups

详细信息	查看全文 | 推荐本文 |

• 作者：Primo啪 Poto膷nik^a ; ^{primoz.potocnik@fmf.uni-lj.si} ; Pablo Spiga^b ; ¹ ; ^{spiga@maths.uwa.edu.au} ; Gabriel Verret^a ; ^{gabriel.verret@fmf.uni-lj.si}
• 关键词：Arc-transitive graphs ; Semiprimitive group ; Primitive group ; Graph-restrictive group ; Weiss Conjecture
• 刊名：Journal of Combinatorial Theory ; Series B
• 出版年：2012
• 期刊代码：135_00958956
• 类别：cp
• 出版时间：May, 2012
• 卷：102
• 期：3
• 页码：820-831
• 文件大小：198 K

摘要

Let 螕 be a connected G-vertex-transitive graph, let v be a vertex of 螕 and let be the permutation group induced by the action of the vertex-stabiliser on the neighbourhood . Then is said to be locally-L. A transitive permutation group L is graph-restrictive if there exists a constant such that, for every locally-L pair and an arc of 螕, the inequality holds.

Using this terminology, the Weiss Conjecture says that primitive groups are graph-restrictive. We propose a very strong generalisation of this conjecture: a group is graph-restrictive if and only if it is semiprimitive. (A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either transitive or semiregular.) Our main result is a proof of one of the two implications of this conjecture, namely that graph-restrictive groups are semiprimitive. We also collect the known results and prove some new ones regarding the other implication.