Cyclic $m$-DCI-groups and $m$-CI-groups

TL;DR

The paper resolves the classification of cyclic $m$-DCI- and $m$-CI-groups by deriving precise divisibility conditions on $n$ relative to $m$, notably that for $m\ge3$, $\mathbb{Z}_n$ is an $m$-DCI-group iff $n$ is not divisible by $8$ nor by $p^2$ for any odd prime $p<m$, and for $m\ge6$, $\mathbb{Z}_n$ is an $m$-CI-group iff either $n\in\{8,9,18\}$ or $n\notin\{8,9,18\}$ and it is not divisible by $8$ nor by $p^2$ for any odd prime $p<(m-1)/2$. The results synthesize and strengthen earlier sufficient conditions (Dobson) with established necessary conditions, using Muzychuk's criterion to reduce isomorphism problems to explicit permutations (generalized multipliers) acting on key partitions; the outcome provides a complete, explicit boundary between $m$-CI and full CI behavior for cyclic groups, with broader implications for circulant Cayley digraphs and graphs. The approach combines a careful structural analysis of circulant isomorphisms with a reduction to connected subgroups and targeted constructions of non-CI examples to establish necessity.

Abstract

Based on the earlier work of Li (European J. Combin. 1997) and Dobson (Discrete Math. 2008), in this paper we complete the classification of cyclic $m$-DCI-groups and $m$-CI-groups. For a positive integer $m$ such that $m \ge 3$, we show that the group $\mathbb{Z}_n$ is an $m$-DCI-group if and only if $n$ is not divisible by $8$ nor by $p^2$ for any odd prime $p < m$. Furthermore, if $m \ge 6$, then we show that $\mathbb{Z}_n$ is an $m$-CI-group if and only if either $n \in \{ 8, 9, 18 \}$, or $n \notin \{ 8, 9, 18 \}$ and $n$ is not divisible by $8$ nor by $p^2$ for any odd prime $p < \frac{m - 1}{2}$.

Theorems & Definitions (31)

• Theorem 1.1: M97
• Theorem 1.2: L02
• Theorem 1.3: D
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Definition 2.1
• Definition 2.2
• Definition 2.3