On CI-property of normal Cayley digraphs over abelian groups
Grigory Ryabov
TL;DR
This work advances the CI-analysis of normal Cayley digraphs by developing a unifying $S$-ring framework that reduces the problem to abelian $p$-groups. It delivers a complete description of abelian $\text{NCI}^{(2)}$-groups, including a full classification for abelian $2$-groups and explicit odd-order cases, and provides a general criterion ensuring abelian groups with restricted Sylow subgroups are $\text{NCI}^{(2)}$. The methodology hinges on tensor and generalized wreath product decompositions of $p$-S-rings and the Babai-style transjugacy criterion, enabling a reduction from global to Sylow-p analyses. The results yield infinite families of abelian $\text{NCI}^{(2)}$-groups and sharpen understanding of CI-properties in normal Cayley graphs/digraphs, with potential algorithmic implications for isomorphism testing in this class.
Abstract
A Cayley digraph $Γ$ over a finite group $G$ is said to be CI if for every Cayley digraph $Γ^\prime$ over $G$ isomorphic to $Γ$, there is an isomorphism from $Γ$ to $Γ^\prime$ 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 $Γ$ that the group $G_r$ of all right translations of $G$ is normal in $Aut(Γ)$. 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.