Dihedral groups of order $2pq$ or $2pqr$ are DCI

Abstract

A group has the (D)CI ((Directed) Cayley Isomorphism) property, or more commonly is a (D)CI group, if any two Cayley (di)graphs on the group are isomorphic via a group automorphism. That is, $G$ is a (D)CI group if whenever $\mathrm{Cay}(\mathrm{G},\mathrm{S})\cong \mathrm{Cay}(\mathrm{G},\mathrm{T})$, there is some $\delta \in \mathrm{Aut}(\mathrm{G})$ such that $S^{\delta }=T$. (For the CI property, we only require this to be true if $S$ and $T$ are closed under inversion.) Suppose $p,q,r$ are distinct odd primes. We show that $D_{2pqr}$ is a DCI group. We present this result in the more general context of dihedral groups of squarefree order; some of our results apply to any such group, and may be useful in future toward showing that all dihedral groups of squarefree order are DCI groups.

arXiv comment31 pages