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A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if for every $Tsubseteq G$, $Cay(G,S)cong Cay(G,T)$ implies that there is $sigmain Aut(G)$ such that $S^sigma=T$. We call a group $G$ a NDCI-group if all normal Cayley digraphs of $G$ are CI-digraphs, and a NCI-group if all normal Cayley graphs of $G$ are CI-graphs, respectively. In this paper, we prove that a cyclic group of order $n$ is a NDCI-group if and only if $8 mid n$, and is a NCI-group if and only if either $n=8$ or $8 mid n$.
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $alphain Aut(G)$ such that $S
A Cayley graph is said to be an NNN-graph if it is both normal and non-normal for isomorphic regular groups, and a group has the NNN-property if there exists an NNN-graph for it. In this paper we investigate the NNN-property of cyclic groups, and show that cyclic groups do not have the NNN-property.
In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~ref{111015a} where a short li
In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if $R$ is a generalised dihedral group and if $R$ is a CI-group, then for every odd prime $p$ the Sylow $p$-subgroup of $R$ has order $p$, or $9$. Consequent
In this paper, we construct an infinite family of normal Cayley graphs, which are $2$-distance-transitive but neither distance-transitive nor $2$-arc-transitive. This answers a question raised by Chen, Jin and Li in 2019 and corrects a claim in a literature given by Pan, Huang and Liu in 2015.