On isomorphisms of Cayley digraphs on dihedral groups

In this paper, we investigate m-DCI and m-CI properties of dihedral groups. We show that for any m E {I, 2, 3}, the dihedral group D2k is m-DCI if and only if D2k is m-CI if and only if 2 f k. § 1. Preliminaries Let G be a finite group and 5 a subset of G with 1 1:. 5. We use r = Cay( G; 5) to denote the Cayley digraph of G with respect to 5, defined to be the directed graph with vertex set and edge set given by v(r) = G, E(r) = {(g,8g) 1 9 E G, 8 E 5}. When a digraph contains both undirected edges and directed edges, we refer to directed edges as arcs and undirected edges as edges. Let D2k be the dihedral group, D2k = (0;,,131 o;k = 1, (3a{3 = 0;-1). Whenever we refer to 0; and {3 in this paper, we mean the generators of the dihedral group D2k . *The work for this paper was supported by the National Natural Science Foundation of China and the Doctorial Program Foundation of Institutions of Higher Education (P. R. China). Australasian Journal of Combinatorics 15(1997), pp.213-220 We use ord(g) to denote the order of an element 9 in a group, use 151 to denote the cardinal number of a set 5, and use gcd(i,j) to denote the greatest common divisor of two integers i and j. Let Cay( G; 5) be the Cayley digraph of G with respect to 5. Take 7r E Aut( G) and set S7r T. Obviously we have Cay( G; 5) ~ Cay( G; T). This kind of isomorphism between two Cayley digraphs is called a Cayley isomorphism. DEFINITION 1.1 Given a subset 5 of G, we call 5 a CI-subset of G, if for any subset T of G with Cay( G; 5) ~ Cay( G; T), there exists 7r E Aut( G) such that 5 = T. DEFINITION 1.2 A finite group G is called an m-DCI-group if any subset 5 ofG with 1 t/:. 5 and 151 ::; m is a CI-subset. The group G is called an m-CI-group if any subset 5 ofG with 1 t/:. 5,5-1 = 5 and 151 ::; m is a CI-subset, where 51 = {S-l Is E 5}. A number of authors have investigated the m-DCI properties of abelian groups for m ::; 3, and m-CI properties of abelian groups for m ::; 5 (see [1-6]). THEOREM 1.3 ([7, Theorem 2.5], or see [1-6]) 1. The finite cyclic group Zk is m-DCI if2 t k, m = 1,2,3. 2. Any finite cyclic group Zk is 4-C1. DEFINITION 1.4 A finite group G is called homogeneous if whenever Hand f{ are two isomorphic subgroups and (J is an isomorphism (J : H -+ K, then (J can be extended to an automorphism of G. The following lemmas are very easy to prove. LEMMA 1.5 IfCay(G;5) ~ Cay(G;T), then 1(5)1 = I(T)I. LEMMA 1.6 1. Any finite cyclic group Zk is homogeneous. 2. For any finite cyclic group Zk and for any a, b E Zk with ord( a) = ord(b), there exists an 7r E Aut(Zk) such that a7r = b. LEMMA 1.7 For any i E Z set, = cif3. Then D2k (cx" I cxk = ,2 = 1, ,cx, = -1) cx . LEMMA 1.8 Given 7r E Aut(Zk) and x E Zk. Define a mapping 7r : D2k --+ D2k by Then 7r E Aut(D2k ). LEMMA 1.9 The dihedral group D2k is homogeneous if 2 t k.