A nonabelian CI-group

In this note, we prove that the alternating group A4 is a CI-group and that all disconnected Cayley graphs of A5 are CI-graphs. As a corollary, we conclude that there are exactly 22 non-isomorphic Cayley graphs of A4 • Let G be a finite group and set G# = G \ {I}. For a subset S ~ G# with S = S-l := {S-l Is E S}, the Cayley graph is the graph Cay(G, S) with vertex set G and with x and y adjacent if and only if yxE S. For an automorphism ()" of G, it easily follows that Cay(G, S) 9E Cay(G, S'"), The graph Cay(G, S) is called a CI-graph of G if, whenever Cay(G, S) 9E Cay(G, T), there is an element ()" E Aut(G) such that So= T (CI stands for Cayley Isomorphism). A finite group G is called a CI-group if all Cayley graphs of G are CI-graphs. Adam (1967) conjectured that all finite cyclic groups were CI-groups, and this conjecture was disproved by Elspas and Turner (1970). Since then, a lot of work has been devoted to seeking CI-groups in the literature (see for example [2, 3, 9]). So far, the known CI-groups are the following groups: (1) Zs, Zg, Zn, Z2n and Z4n, where n is odd square-free, see [9, 10]; (2) Qs, Z;, Z~, where p is a prime, see [3, 5, ll]; (3) D2p , F3p (the Frobenius group of order 3p), where p is a prime, see [2, 6]. Recently, an explicit list of groups which contains all finite CI-groups was produced by C. E. Praeger and the second author in [8] (also see [6]). Unfortunately, even with this knowledge, it is still a very hard problem to obtain a complete classification of finite CI-groups. By [6], the candidates of indecomposable CI-groups may be divided into three classes, two of them consist of infinite families, and the other contains 9 "sporadic" groups: Zs, Zg, Zg ><l Zg, Q8, Australasian Journal of Combinatorics 17(1998), pp.229-233 Q8 ~ Z3 and Q8 ~ Zg. With the assistance of computer, B. D. McKay determined cyclic CI-groups of order at most 37, and in particular proved that Za and Zg are CI-groups (unpublished). By [11], Qa is a CI-group. In this note, we shall prove that Z~ ~ Z3 (~ A4) is a CI-group. However, it is not known which of the other sporadic candidates (that is, Zg ~ Z2, Zg ~ Z4, Z~ ~ Zg, Q8 ~ Z3 and Q8 ~ Zg) are CI-groups. One of interests of studying CI-groups is to classify Cayley graphs of the corresponding groups. As an application of the result that A4 is a CI-group, we shall give a classification of Cayley graphs of A4. For a positive integer m, a group G is called an m-CI-group if all Cayley graphs of G of valency at most mare CI-graphs. In [7], it is proved that a nonabelian simple group is a 3-CI-group if and only if it is As. However, it is still an open question whether As is a 4-CI-group (see [7]). We shall prove that all disconnected Cayley graphs of As are CI-graphs. Theorem 1 The alternating group A4 of order 12 is a CI-group. Proof. For a positive integer m, a group G is said to have the m-CI property if all Cayley graphs of G of valency mare CI-graphs. Clearly, the group G has the m-CI property if and only if G has the (IG#I m)-CI property. Therefore, to prove that A4 is a CI-group, we only need to prove that A4 has the m-CI property for m ::; 5. Let G = A4 ~ Z~ ~ Z3. Then G contains three involutions (elements of order 2): aI, a2, a3, and four subgroups of order 3: (Xi) (i = 1,2,3,4). Let Sy13(G) be the set of Sylow 3-subgroups of G. It is easily checked that the following properties are true: (a) (Xi) acts (by conjugation) transitively on the set {al,a2,a3}; (b) Z~ = {I, all a2, a3} acts (by conjugation) regularly on the set SyI3(G); (c) G acts (by conjugation) 2-transitively on Sy13(G). (If a Cayley graph Cay(G, 8) is a CI-graph, S is called a CI-subset.) By (a), we know that G has the 1-CI property. By (a) and (b), it follows that G has the 2-CI property. Let 8 be a subset of G# of size 3 with S = S-1. If 8 consists of three involutions, then (8) Z~. Noting that G has the unique subgroup (S) of order 4, 8 contains all the involutions of G. For any T E G# such that Cay(G,8) ~ Cay(G,T), we have that I (S) I = I (T) I and so S = T. Thus 8 is a CI-subset. Let 8 = {x, x-I, a} and T {x', X,-1, al}, where o(x) = o(x') = 3 and o(a) = o(a') = 2, such that Cay(G,S) ~ Cay(G,T). Since (x) is conjugate to (x'), there exists an element y E G such that {x,x-1}y = {XI,XII }, so SY = {x',x'-\aY }. Further, as (Xl) is transitive on the set of all involutions of G, tl.tere exists an integer j such that x,j maps aY to a', and so 8 yx'J = {x',x'-\ayy'J = {xl,xl-l,a'} = T. Thus 8 is a CI-subset, and so G has the 3-CI property. (For s E 8, an edge {u, v} of Cay(G, 8) is called an s-edge if vu-1 = s.) Let S be a subset of G# of size 4 with 8 = S-1, and let Rl = {al,a2,x1,x1 } and R2 = {Xl,X1\X2,X2 }.