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理学院2018-2019学年秋季第十三周学术报告(3)


学 术 报 告


报告题目:On Automorphisms of Haar graphs of Abelian Groups


报告学者:Ted Dobson 教授


报告者单位斯洛文尼亚, The University of Primorska

 

报告时间2018年12月5日(周三)上午11:00-12:00


报告地点学生活动中心10层1005


报告摘要: Let $G$ be a group and $S\subseteq G$.  A Haar graph of $G$ with connection set $S$ has vertex set $\Z_2\times G$ and edge set $\{(0,g)(1,gs):g\in G{\rm\ and\ }s\in S\}$.  Haar graphs are then natural bipartite analogues of Cayley digraphs.  We first examine the relationship between the automorphism group of a Cayley digraph of $G$ with connection set $S$ and a Haar graph of $G$ with connection set $S$.  We establish that the automorphism group of a Haar graph contains a natural subgroup isomorphic to the automorphism group of the corresponding Cayley digraph.  In the case where $G$ is abelian, we then give four situations in which the automorphism group of the Haar graph can be larger than the natural subgroup corresponding to the automorphism group of the Cayley digraph together with a specific involution, and analyze the full automorphism group in each of these cases. As an application, we show that all $s$-transitive Cayley graphs of generalized dihedral groups have a quasiprimitive automorphism group, can be ``reduced" to $s$-arc-transitive graphs of smaller order, or are Haar graphs of abelian groups whose automorphism groups have a particular permutation group theoretic property.


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