标题： 关于交换群上正规Cayley有向图的CI性质 显示英文标题

标题： On CI-property of normal Cayley digraphs over abelian groups

摘要： 一个有限群$G$上的Cayley有向图$\Gamma$被称为CI，如果对于每一个与$\Gamma$同构的$G$上的Cayley有向图$\Gamma^\prime$，存在一个从$\Gamma$到$\Gamma^\prime$的同构，该同构同时是$G$的自同构。 在本文中，我们研究了阿贝尔群上正规Cayley有向图的CI性质，即这样的Cayley有向图$\Gamma$，其中群$G_r$是$G$的所有右平移的群，在$Aut(\Gamma)$中是正规的。首先，我们将任意阿贝尔群的情况简化为阿贝尔$p$-群的情况。进一步地，我们得到了关于阿贝尔$p$-群上正规Cayley有向图的CI性质的几个结果。 特别地，我们证明了每个阶数至多为$p^5$的交换$p$-群上的正规 Cayley 有向图，其中$p$是一个奇素数，都是 CI。 显示英文摘要

摘要： A Cayley digraph $\Gamma$ over a finite group $G$ is said to be CI if for every Cayley digraph $\Gamma^\prime$ over $G$ isomorphic to $\Gamma$, there is an isomorphism from $\Gamma$ to $\Gamma^\prime$ which is at the same time an automorphism of $G$. In the present paper, we study a CI-property of normal Cayley digraphs over abelian groups, i.e. such Cayley digraphs $\Gamma$ that the group $G_r$ of all right translations of $G$ is normal in $Aut(\Gamma)$. At first, we reduce the case of an arbitrary abelian group to the case of an abelian $p$-group. Further, we obtain several results on CI-property of normal Cayley digraphs over abelian $p$-groups. In particular, we prove that every normal Cayley digraph over an abelian $p$-group of order at most $p^5$, where $p$ is an odd prime, is CI.