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

TL;DR

This paper completes the classification of cyclic $m$-DCI-groups and $m$-CI-groups, providing necessary and sufficient conditions based on divisibility properties of the group order.

Contribution

It offers a complete characterization of cyclic $m$-DCI and $m$-CI groups for all relevant $m$, extending previous partial results.

Findings

$ ext{Z}_n$ is an $m$-DCI-group iff $n$ not divisible by 8 or $p^2$ for odd primes $p<m$

$ ext{Z}_n$ is an $m$-CI-group iff $n otin ext{set}\{8,9,18\}$ and not divisible by 8 or $p^2$ for odd primes $p<rac{m-1}{2}$ (for $m ext{ } ext{ge } 6$)

Complete classification of cyclic $m$-DCI and $m$-CI groups for $m ext{ } ext{ge } 3$

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 $p < \frac{m - 1}{2}$.