Dihedral groups with the m-DCI property

2022/10/21 by Jin-Hua Xie, Xie, Jin-Hua, Yan‐Quan Feng +3 · 3 citations
Chemistry · Engineering · Mathematics · #05C25 #20B15 #Carbohydrate Chemistry and Synthesis #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2210.11700

openalex publication_date 2022/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A Cayley digraph \rmCay(G,S) of a group G with respect to a subset S of G is called a CI-digraph if for any Cayley digraph \rmCay(G,T) isomorphic to \rmCay(G,S), there is an α∈ \rmAut(G) such that S^α=T. For a positive integer m, G is said to have the m-DCI property if all Cayley digraphs of G with out-valency m are CI-digraphs. Li [The Cyclic groups with the m-DCI Property, European J. Combin. 18 (1997) 655-665] characterized cyclic groups with the m-DCI property, and in this paper, we characterize dihedral groups with the m-DCI property. For a dihedral group D_2n of order 2n, assume that D_2n has the m-DCI property for some 1 ≤ m≤ n-1. Then it is shown that n is odd, and if further p+1≤ m≤ n-1 for an odd prime divisor p of n, then p²\nmid n. Furthermore, if n is a power of a prime q, then D_2n has the m-DCI property if and only if either n=q, or q is odd and 1≤ m≤ q.

Cited by

• Dihedral groups of square-free order are DCI-groups
• Enumerating Cayley digraphs on dihedral groups
• On kernel isomorphisms of m-Cayley digraphs and finite 2PCI-groups

Related

• Dihedral groups of order 2pq or 2pqr are DCI
• On Isomorphisms of Tetravalent Cayley Digraphs over Dihedral Groups
• Normal Cayley digraphs of dihedral groups with CI-property
• Generalized quaternion groups with the m-DCI property
• On isomorphisms of m-Cayley digraphs