Normal Cayley digraphs of cyclic groups with CI-property
aa r X i v : . [ m a t h . C O ] F e b NORMAL CAYLEY DIGRAPHS OF CYCLIC GROUPS WITHCI-PROPERTY
JIN-HUA XIE, YAN-QUAN FENG, YING-LONG LIU
Abstract.
A Cayley (di)graph Cay(
G, S ) of a group G with respect to a subset S of G is called normal if the right regular representation of G is a normal subgroup in thefull automorphism group of Cay( G, S ), and is called a CI-(di)graph if for any Cayley(di)graph Cay(
G, T ), Cay(
G, S ) ∼ = Cay( G, T ) implies that there is σ ∈ Aut( G ) suchthat S σ = T . A group G is called a DCI-group or a NDCI-group if all Cayley digraphsor normal Cayley digraphs of G are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of G are CI-graphs, respectively.DCI-groups and CI-groups have been widely studied in the literature. A lot ofefforts had been made to attach the classifications of cyclic DCI-groups and CI-groups,motivated by the ´Ad´am’s conjecture, and finally Muzychuk finished the classificationsby proving that a cyclic group of order n is a DCI-group if and only if n = mk where m = 1 , , k is odd square-free, and is a CI-group if and only if either n = 8 , , n = mk where m = 1 , , k is odd square-free. In this paper we prove that acyclic group of order n is a NDCI-group if and only if 8 ∤ n , and is a NCI-group if andonly if either n = 8 or 8 ∤ n . k eywords. Cayley digraph, DCI-group, CI-group, NDCI-group, NCI-group. Introduction
A pair of sets (
V, E ) is called a digraph if E ⊆ V × V , and V and E are the vertexset and arc set of the digraph, respectively. A digraph ( V, E ) is called a graph if E issymmetric, that is, E = E − := { ( u, v ) | ( v, u ) ∈ E } , and in this case, an edge { u, v } in the graph corresponds to two arcs ( u, v ) and ( v, u ). Let Γ be a (di)graph. Denote by V (Γ), E (Γ), Arc(Γ) and Aut(Γ) the vertex set, edge set, arc set, and full automorphismgroup of Γ, respectively. All (di)graphs considered in this paper are finite and simple,that is, no multiple arcs or edges between two given vertices, and all groups are finite.Let G be a finite group and S a subset of G . The Cayley digraph
Cay(
G, S ) of G withrespect to S is defined to be the digraph with vertex set G and arc set { ( g, sg ) | g ∈ G, s ∈ S } , and in this case, we say that S is a Cayley subset of G . For a given g ∈ G ,the right multiplication R ( g ) is a permutation on G such that x R ( g ) = xg for any x ∈ G .Then the permutation group R ( G ) = { R ( g ) | g ∈ G } on G is called the right regularrepresentation of G , and it is a regular subgroup of Aut(Cay( G, S )), so that Cay(
G, S ) isvertex-transitive. In fact, Sabidussi [38] proved that a digraph Γ is a Cayley digraph of agroup G if and only if Aut(Γ) contains a regular subgroup isomorphic to G . Furthermore,a Cayley digraph Cay( G, S ) has out-valency | S | , and is connected if and only if h S i = G ,namely, S generates G . Mathematics Subject Classification.
A Cayley digraph Cay(
G, S ) is called a
Cayley graph of G with respect to S if S isan inverse-closed set, that is, S = S − := { x − | x ∈ S } . In this case, Cay( G, S ) canbe viewed as an undirected graph by identifying two arcs ( u, v ) and ( v, u ) as an edge { u, v } . Thus, Cayley graph is a special case of Cayley digraph.Let G be a group. Two Cayley (di)graphs Cay( G, S ) and Cay(
G, T ) of G are calledCayley isomorphic if S σ = T for some σ ∈ Aut( G ). It is easy to see that Cayleyisomorphic Cayley (di)graphs are isomorphic as (di)graphs. However, the converse isnot true, and there are isomorphic Cayley (di)graphs which are not Cayley isomorphic.A Cayley subset S of G is called a CI-subset if for any T ⊆ G , Cay( G, S ) ∼ = Cay( G, T )implies that they are Cayley isomorphic, that is, T = S σ for some σ ∈ Aut( G ), and inthis case, we call Cay( G, S ) a
CI-graph , where CI represents Cayley isomorphic. We saythat a group G is a DCI-group or a
CI-group if all Cayley digraphs or graphs of G areCI-graphs, respectively.A lot of efforts have been made to attach classifications of DCI-groups and CI-groups.We do not intend to give a full account on their study in general, and interested readeris referred to [40, 41, 22, 31, 4, 26, 28, 27, 34, 39], where many authors like Li, Kov´acs,Spiga et al. gave a lot of contributions. Here we only outline classifications of cyclicDCI-groups and CI-groups, which originated from a conjecture proposed by ´Ad´am [1]:every finite cyclic group is a CI-group. Though the conjecture was disproved by Elspasand Turner [15], it started the study of classifications of cyclic DCI-groups and CI-groups, which had been lasted for 30 years. It was known that the ´Ad´am’s conjecture istrue for the cyclic group Z n in each of the following cases ( p and q are distinct primes): n = p [8, 15, 42]; n = 2 p [5]; n = pq [3, 24]; n = 4 p with p > n, φ ( n )) = 1 [35]where φ is the Euler’s function. Finally, Muzychuk [32, 33] put the last piece into thepuzzle and proved that a cyclic group of order n is a DCI-group if and only if n = mk where m = 1 , , k is odd square-free, and is a CI-group if and only if either n = 8 , ,
18, or n = mk where m = 1 , , k is odd square-free. This is one of theremarkable achievements on the study of DCI-groups and CI-groups.A Cayley (di)graph Cay( G, S ) is called normal if the right regular representation R ( G )of G is a normal subgroup of Aut(Cay( G, S )). Godsil [25] proved that if Cay(
G, S ) isnormal then Aut(Cay(
G, S )) is a semidirect product of R ( G ) by the subgroup Aut( G, S )(also see Proposition 2.1), and hence Aut(Cay(
G, S )) is completely determined by Aut( G ).Wang et al. [43] obtained all disconnected normal Cayley graphs. The normality ofCayley graphs of cyclic groups of order a prime and of groups of order twice a primewas solved by Alspach [2] and Du et al. [14], respectively. Dobson [9] determined allnon-normal Cayley graphs of order a product of two distinct primes, and Dobson andWitte [10] determined all non-normal Cayley graphs of order a prime-square. For nor-mality of Cayley graphs of finite simple groups, we refer the reader to [29, 20, 16, 17,18, 45, 46, 19, 13, 12, 36, 47], and for some results on normality of Cayley digraphs,one may see [6, 23, 21, 48]. Based on these results, Xu [44] conjectured that almost allconnected Cayley graphs are normal.A group G is called a NDCI-group or a
NCI-group if all normal Cayley digraphs orgraphs of G are CI-graphs, respectively. Obviously, a DCI-group is a NDCI-group anda CI-group is a NCI-group. Li [30] constructed some normal Cayley digraphs of cyclicgroups of order 2-power which are not CI-graphs, and proposed the following problem:Characterize normal Cayley digraphs which are not CI-graphs. Motivated in part by ORMAL CAYLEY DIGRAPHS OF CYCLIC GROUPS WITH CI-PROPERTY 3 this and classifications of cyclic DCI-groups and CI-groups, we would like to propose thefollowing problem.
Problem 1.1.
Classify finite NDCI-groups and NCI-groups.
In this paper, we solve the above problem for cyclic groups.
Theorem 1.2.
A cyclic group Z n of order n is a NDCI-group if and only if 8 ∤ n . Corollary 1.3.
A cyclic group Z n of order n is a NCI-group if and only if either n = 8or 8 ∤ n .All the notation and terminologies used throughout this paper are standard. Forgroup and graph theoretic concepts which are not defined here we refer the reader to[7, 37, 11]. 2. Preliminaries
In this section, we give some basic concepts and facts that will be needed later. LetCay(
G, S ) be a Cayley digraph of a group G with respect to S . Write Aut( G, S ) = { α ∈ Aut( G ) | S α = S } . It is easy to see that Aut( G, S ) is a subgroup of Aut(Cay(
G, S )),and in fact, it is a subgroup of the stabilizer (Aut(Cay(
G, S ))) of 1 in Aut(Cay( G, S )).Godsil [25] proved that the normalizer of R ( G ) in Aut(Cay( G, S )) is a semiproduct of R ( G ) by Aut( G, S ), and then one may prove the following proposition easily (see [44,Propositions 1.3 and 1.5]).
Proposition 2.1.
Let Cay(
G, S ) be a Cayley digraph of a group G with respect to S and let A = Aut(Cay( G, S )). Then N A ( R ( G )) = R ( G ) ⋊ Aut(
G, S ) and Cay(
G, S ) isnormal if and only if A = Aut( G, S ).Babai [5] obtained the well-known criterion for a Cayley digraph to be a CI-digraph:a Cayley digraph Cay(
G, S ) is a CI-digraph if and only if all regular subgroups ofAut(Cay(
G, S )) isomorphic to G are conjugate. This was also proved by Alspach andParsons [3]. Based on this criterion, the follow proposition is straightforward (also see[30, Corollary 6.9]). Proposition 2.2.
Let Cay(
G, S ) be a normal Cayley digraph of a group G with respectto S . Then Cay( G, S ) is a CI-digraph if and only if R ( G ) is the unique regular subgroupof Aut(Cay( G, S )) which is isomorphic to G .Li [30, Example 6.10] constructed some normal Cayley digraphs of cyclic groups oforder 2-power which are not CI-digraphs. Proposition 2.3.
Let G = h a i ∼ = Z r with r ≥ S = { a, a , a r − +1 } . ThenCay( G, S ) is a normal Cayley digraph but not a CI-digraph.The classifications of cyclic DCI-groups and CI-groups were finally completed byMuzychuck [32, 33] (also see [30, Theorem 7.1]).
Proposition 2.4.
A cyclic group of order n is a DCI-group if and only if n = k, k or4 k where k is odd square-free, and a cyclic group of order n is a CI-group if and only ifeither n ∈ { , , } or n = k, k or 4 k where k is odd square-free. JIN-HUA XIE, YAN-QUAN FENG, YING-LONG LIU
It is well-known that Aut( Z n ) ∼ = Z × Z n − for n ≥ Z p n ) ∼ = Z ( p − p n − foran odd prime p . Let G be a group and let G i ≤ G for 1 ≤ i ≤ n , with G = G × G ×· · · × G n , that is, G is a direct product of subgroups G , G , · · · , G n . Then Aut( G i ) canbe naturally extended to a group of automorphisms of G : for any α i ∈ Aut( G i ), define( g · · · g i − g i g i +1 · · · g n ) α i = g · · · g i − g α i i g i +1 · · · g n with g j ∈ G j for any 1 ≤ j ≤ n , andfor not making the notation too cumbersome, we still denote by Aut( G i ) this extendedgroup of automorphisms of G . It follows that Aut( G )Aut( G ) · · · Aut( G n ) = Aut( G ) × Aut( G ) × · · · × Aut( G n ) ≤ Aut( G ). Furthermore, if G i is a Sylow subgroup of G foreach 1 ≤ i ≤ n , then G i is characteristic in G and hence Aut( G ) induces a group ofautomorphisms of G i , which implies that Aut( G ) = Aut( G ) × Aut( G ) × · · · × Aut( G n ).In particular, we have the following proposition (see [37, Theorem 7.3]). Proposition 2.5.
For a positive integer n , let n = m Q i =1 p r i i be the prime factorization of n , and let Z n = Z p r × Z p r × · · · × Z p rmm . Then Aut( Z n ) = Aut( Z p r ) × Aut( Z p r ) × · · · × Aut( Z p rmm ). 3. Proof of Theorem 1.2
Let g be an element of a group G , and denote by o ( g ) the order of g in G . We firstprove the following lemma that is crucial in the proof of Theorem 1.2. Lemma 3.1.
For a positive integer n , let n = m Q i =1 p r i i be the prime factorization of n forits distinct prime factors p , p , · · · , p m , and let S be a Cayley subset of the cyclic group Z n . Assume that p t is an odd prime and Aut( Z n , S ) contains an element of order p t inAut( Z p rtt ) for some 1 ≤ t ≤ m . Then Cay( Z n , S ) is a non-normal Cayley digraph. Proof.
Write Z n = Z p r × Z p r × · · · × Z p rmm = h a i × h a i × · · · × h a m i with o ( a i ) = p r i i .Then Z p rii = h a i i , and since p i is a factor of n , we have r i ≥ ≤ i ≤ m .Furthermore, each x ∈ Z n can be uniquely factorized as x = a s a s · · · a s i i · · · a s m m , and wecall a s i i the i -th component of x , denoted by x i , that is, x i = a s i i ∈ h a i i = Z p rii .Since p t is odd, Aut( Z p rtt ) ∼ = Z ( p t − p rt − t and hence Aut( Z p rtt ) has an element of order p t if and only if r t ≥
2: in this case, Aut( Z p rtt ) has a unique subgroup of order p t . SinceAut( Z n , S ) contains an element of order p t in Aut( Z p rtt ), we have r t ≥
2. In fact, anelement δ of order p t in Aut( Z p rtt ) can be defined by( a j a j . . . a j t − t − a j t t a j t +1 t +1 . . . a j m m ) δ = a j a j . . . a j t − t − ( a j t t ) p rt − t +1 a j t +1 t +1 . . . a j m m for any a j k k ∈ Z p rkk , or alternatively by x δ = x ( x t ) p rt − t for any x ∈ Z n . The uniqueness of subgroup of order p t in Aut( Z p rtt ) implies that h δ i ≤ Aut( Z n , S ).Let M = h a , a , · · · , a t − , a p t t , a t +1 , · · · , a m i , and H = h a p rt − t t i . Then | Z n : M | = p t and Z n = S p t − i =0 a it M , a coset decomposition of M in Z n . Further-more, | H | = p t , H ≤ Z p rtt ≤ Z n , and H is the unique subgroup of order p t in Z p rtt . Since ORMAL CAYLEY DIGRAPHS OF CYCLIC GROUPS WITH CI-PROPERTY 5 r t ≥ M also has a unique subgroup of order p t , which must be H . It follows that H ≤ M ≤ Z n . Claim 1: δ fixes each coset of H in Z n setwise and M is the fixed-point set of δ in Z n .Let x ∈ Z n . Since x δ = x ( x t ) p rt − t and r t ≥ δ fixes H = h a p rt − t t i pointwise, andhence ( xH ) δ = x ( x t ) p rt − t H = xH , that is, δ fixes each coset of H in Z n setwise.If x δ = x then x = x ( x t ) p rt − t , that is, x p rt − t t = 1. Since x t ∈ h a t i , we have x t ∈ h a p t t i ,implying x ∈ M . On the other hand, for any y ∈ M we have y t = a p t s t t and hence y δ = yy p rt − t t = y ( a p t s t t ) p rt − t = y , that is, δ fixes M pointwise. Thus, M is the fixed-pointset M in Z n . This completes the proof of Claim 1.Let Γ = Cay( Z n , S ) and A = Aut(Γ). Recall that h δ i ≤ Aut( Z n , S ) ≤ A , and A isthe stabilizer of 1 in A . For a subset T ⊆ V (Γ), denote by [ T ] the induced sub-digraphof T in Γ, that is, the digraph with the vertex set T and with an arc ( u, v ) ∈ Arc([ T ]), u, v ∈ T , whenever ( u, v ) ∈ Arc(Γ).
Claim 2:
Let α be an automorphism of [ M ] fixing every coset of H in M setwise. Then α can be extended to an automorphism ¯ α of Γ such that ¯ α fixes each vertex in Z n \ M .Take a coset zH in M and a coset yH in Z n \ M . Denote by [ zH, yH ] the inducedbipartite digraph with vertex set zH ∪ yH and all arcs from zH to yH of Γ. To provethe claim, it suffices to show that both [ zH, yH ] and [ yH, zH ] are isomorphic to eitherthe empty graph with 2 p t isolated vertices, or the complete bipartite digraph ~K p t ,p t oforder 2 p t . Note that zM = yM .Let w ∈ zH ⊆ M . By Claim 1, h δ i fixes w and is transitive yH , because δ has order p t and | yH | = p t . Since δ ∈ A , if w has an out-neighbour in yH then every vertex in yH is an out-neighbour of w , and by the transitivity of R ( H ) ≤ A on both zH and yH , every vertex in yH is an out-neighbour of any vertex in zH . Thus, the induceddigraph [ zH, yH ] is isomorphic to either the empty graph with 2 p t isolated vertices, orthe complete bipartite digraph ~K p t ,p t .Note that the choices of zH in M and yH in Z n \ M are arbitrary. Since R ( a t ) ≤ A wehave that for any two cosets uH and vH with uM = vM , the induced digraph [ uH, vH ]is isomorphic to either the empty graph with 2 p t isolated vertices, or the completebipartite digraph ~K p t ,p t . In particular, this is true for [ yH, zH ], as required.To have an intuitive impress for the structure of Γ, assume that [ H, a t H ] ∼ = ~K p t ,p t andthere are no other arcs initiating from H . Since R ( Z n ) ≤ A , we may deduce all arcs inFigure 1, where a line segment with an arrow represents a ~K p t ,p t , and M p t is the uniqueSylow p t -subgroup of M . Let b = a · · · a t − a t +1 · · · a m . We have that M p t = p rt − t − [ k =0 a kp t t H, M = n/p rtt − [ j =0 b j M p t , and Z n = p t − [ i =0 n/p rtt − [ j =0 p rt − t − [ k =0 a it b j a kp t t H. Then we may depict the graph Γ as Figure 1.Since M ≤ Z n , the induced sub-digraph [ M ] of M in Γ is exactly the Cayley digraphCay( M, M ∩ S ), that is, [ M ] = Cay( M, M ∩ S ). Let β = ˆ R ( a p rt − t t ) be the right multi-plication of a p rt − t on M (note that R ( a p rt − t t ) is the right multiplication of a p rt − t on Z n ). JIN-HUA XIE, YAN-QUAN FENG, YING-LONG LIU
H a p t t H m = p r t − t · · · · · · a m − p t t H · · · · · · cH ca p t t H m = p r t − t · · · · · · ca m − p t t H MM p t cM p t with c = b ( n/p rtt ) − a t H a p t +1 t H · · · · · · a m − p t +1 t H · · · · · · ca t H ca p t +1 t H · · · · · · ca m − p t +1 t H a t M ...... ...... ...... ...... a p t − t H · · · · · · a m − p t − t H a m − t H · · · · · · ca p t − t H · · · · · · ca m − p t − t H ca m − t H a p t − t Ma p t − t H · · · · · · a m − p t − t H a m − t H · · · · · · ca p t − t H · · · · · · ca m − p t − t H ca m − t H a p t − t M ❄ ❄ ❄ ❄ ❄ ❄❄ ❄ ❄ ❄ ❄ ❄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✗ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✗❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❪ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✗ ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✗❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❪ ~K p t ,p t Figure 1.
The structure of Γ.Then β ∈ Aut([ M ]). For any coset zH of H in M , we have ( zH ) β = zHa p rt − t t = zH ,that is, β fixes every coset of H in M setwise. By Claim 2, ¯ β ∈ Aut(Γ) fixes everyvertex in Z n \ M and has the same action as β on M . Note that R ( a t ) ∈ Aut(Γ) and M R ( a t ) = a t M . Then ¯ β R ( a t ) ∈ Aut(Γ) fixes each vertex in Z n \ a t M . In particular, ¯ β R ( a t ) fixes 1 as a t M = M , and hence ¯ β R ( a t ) ∈ A .Now we are ready to finish the proof. Suppose to the contrary that Γ is normal. ByProposition 2.1, A = Aut( Z n , S ). It follows ¯ β R ( a t ) ∈ Aut( Z n , S ) and hence ( a t ) ¯ β R ( at ) =( a ¯ β R ( at ) t ) . Since a t M = a t M , we have ( a t ) ¯ β R ( at ) = a t . On the other hand, a ¯ β R ( at ) t = a R ( a − t ) ¯ βR ( a t ) t = 1 ¯ βR ( a t ) = a p rt − t t a t and ( a ¯ β R ( at ) t ) = a t a p rt − t t . It follows that a t = a t a p rt − t t ,that is, a p rt − t t = 1. This implies that p t | o ( a t ) = p r t t , contradicting that p t is odd.Thus, Γ is not normal, as required. (cid:3) Now we construct normal Cayley graphs on cyclic groups which are not CI-graphs.
Lemma 3.2.
Let m and s be positive integers such that s ≥
3, (2 , m ) = 1 and2 s m = 8. Let Z s m = h a i × h b i ∼ = Z s × Z m with o ( a ) = 2 s and o ( b ) = m , and S = { ( ab ) ± , ( a b ) ± , ( a s − +1 b ) ± } . Then Cay( Z s m , S ) is a normal Cayley graph butnot a CI-graph. Proof.
Let Γ = Cay( Z s m , S ) and A = Aut(Γ). Since S = S − , Γ is a graph. Forany u, v ∈ V (Γ), denote by d ( u, v ) the distance between u and v in Γ. Write Γ k ( u ) = { w | d ( u, w ) = k } , the k -step neighborhood of the vertex u in Γ. Then Γ ( u ) = Γ( u ) isthe neighbourhood of u in Γ.Let A be the stabilizer 1 in A , and let A ∗ = { α ∈ A | u α = u for any u ∈ Γ (1) = S } ,namely, the subgroup of A fixing S pointwise. Define α to be the automorphism of Z s m ORMAL CAYLEY DIGRAPHS OF CYCLIC GROUPS WITH CI-PROPERTY 7 induced by a a − and b b − , and β to be the automorphism of Z s m induced by a a s − +1 and b b . It is easy to check that Z × Z ∼ = h α, β i ≤ Aut(
G, S ) ≤ A .Clearly, Γ (1) = { ( a b ) ± , ( a b ) ± , ( a s − +3 b ) ± , ( a s − +2 b ) ± , a s − } . Since s ≥ , m ) = 1 and 2 s m = 8, we derive that | Γ (1) | = 9, and the induced subgroup [Γ (1) ∪ Γ (1) ∪ Γ (1)] can be depicted as Figure 2. tt t t tt tt t t t t t t t t ❍❍❍❍❏❏❏❏❏❏❏❩❩❩❩❩❩❩❩❍❍❍❍❍❍❍❍PPPPPPPPPPPP❏❏❏❏❏❏❏❩❩❩❩❩❩❩❩ ✡✡✡✡✡✡✡ ✚✚✚✚✚✚✚✚ ✟✟✟✟✟✟✟✟ ✏✏✏✏✏✏✏✏✏✏✏✏ ✟✟✟✟ ✟✟✟✟✟✟✟✟✟✟✟✟ ✡✡✡✡✡✡✡ ✚✚✚✚✚✚✚✚ a b a s − +1 b a s − − b − a − b − ab a − b − a b a b a s − +3 b a s − +2 b a s − a s − − b − a s − − b − a − b − a − b − Figure 2.
The induced subgroup [Γ (1) ∪ Γ (1) ∪ Γ (1)] in Γ.Note that A ∗ fixes 1 and S pointwise. For any two distinct vertices u, v ∈ Γ (1), fromFigure 2 we have that Γ ( u ) ∩ S = Γ ( v ) ∩ S , that is, any two distinct vertices in Γ (1)cannot have the same neighbours in Γ (1). Since A ∗ fixes Γ (1) ∪ Γ (1) pointwise andΓ (1) setwise, A ∗ fixes u and v and hence A ∗ fixes Γ (1) pointwise. By the transitivityof A on V (Γ), A ∗ w fixes Γ ( w ) pointwise, for any w ∈ V (Γ). Since h S i = Z s m , Γis connected, and an easy inductive argument on k gives rise to that A ∗ fixes Γ k (1)pointwise for any positive integer k . Thus, A ∗ fixes each vertex of Γ, that is, A ∗ = 1,and in particular, A acts faithfully on S .Again by Figure 2, the induced subgraph [ S ] is a cycle of length 6 and hence Aut([ S ]) ∼ = D , the dihedral group of order 12. Since A is faithful on S , we may assume that A ≤ Aut([ S ]), and since Z × Z ∼ = h α, β i ≤ Aut(
G, S ) ≤ A , we have either A =Aut([ S ]) ∼ = D , or A = Aut( G, S ).Suppose that A = Aut([ S ]) ∼ = D . Then A has a unique Sylow 3-subgroup of order3, and since [ S ] is a cycle of length 6, the unique Sylow 3-subgroup can be generatedby an element of order 3, say γ , such that ( ab ) γ = a s − +1 b , ( a s − +1 b ) γ = a − b − and ( a − b − ) γ = ab . From Figure 2, a s − is the unique vertex in Γ (1) that has fourneighbours in Γ (1), which yields that A fixes a s − , namely, ( a s − ) γ = a s − . Since { ab, a s − } ∈ E (Γ), we have { ab, a s − } γ − ∈ E (Γ), that is, { a − b − , a s − } ∈ E (Γ), whichis not true by Figure 2.It follows that A = Aut( G, S ). By Proposition 2.1, Γ is normal. To finish the proof,we are left to show that Γ is not a CI-graph.Let H = h R ( ab ) β i . Since β − R ( ab ) β = R (( ab ) β ) = R ( a s − +1 b ) and β = 1, we have( R ( ab ) β ) = R ( ab ) R ( ab ) β = R ( ab ) R ( a s − +1 b ) = R ( a s − +2 b ), and since o ( a s − +2 b ) =2 s − m , we obtain that o ( R ( ab ) β ) is even and o ( R ( ab ) β ) = (2 , o ( R ( ab ) β )) o (( R ( ab ) β ) ) =2 s m , that is, H ∼ = Z s m . Note that h R ( a s − +2 b ) i is semiregular on V (Γ) with twoorbits of length 2 s − m , that is, h a s − +2 b i = h a b i and a h a b i . Since 1 R ( ab ) β = ( ab ) β = JIN-HUA XIE, YAN-QUAN FENG, YING-LONG LIU a s − +1 b ∈ a h a b i and h R ( a s − +2 b ) i is a normal subgroup of index 2 in H , R ( ab ) β interchanges the two orbits h a b i and a h a b i , implying that H is transitive on V (Γ).Since | H | = | V (Γ) | , H is regular on V (Γ), and since R ( Z s m ) = H , Γ is not a CI-graphby Proposition 2.2. (cid:3) Let G be a finite group and let p be a prime. Denote by G p a Sylow p -subgroupof G . The right multiplication R ( G ) and the automorphism group Aut( G ) of G arepermutation groups on G . It is easy to see that R ( g ) α = R ( g α ) for any g ∈ G and α ∈ Aut( G ), namely, R ( G ) is normalized by Aut( G ). Furthermore, R ( G ) ∩ Aut( G ) = 1,and hence we have R ( G )Aut( G ) = R ( G ) ⋊ Aut( G ). This group is called the holomorph of the group G , denoted by Hol( G ), that is, Hol( G ) = h R ( G ) , Aut( G ) i = R ( G ) ⋊ Aut( G )(see [37, Lemma 7.16]).Now we are ready to prove Theorem 1.2. Proof of Theorem 1.2:
Let Z n be the cyclic group of order n and let 8 | n . ByProposition 2.3 and Theorem 3.2, there exists a normal non-CI Cayley digraph on Z n ,that is, Z n is not a NDCI-group. The necessity of the theorem follows. To prove thesufficiency, assume that 8 ∤ n and we only need to prove that Z n is a NDCI-group.For convenience, write n = p r p r · · · p r m m s as the prime factorization of n , where s ≤ p , p , · · · , p m are distinct odd prime factors of n such that p > p > · · · > p m . ByProposition 2.4, Z n is a DCI-group for n = 1 , ,
4, and so a NDCI-group. Thus, we mayassume that m ≥ r r · · · r m ≥ Z n , S ) be a normal Cayley digraph and let A = Aut(Γ). To prove that Z n is a NDCI-group, it suffices to show that Γ is a CI-digraph.Let Z n = Z p r × Z p r × · · · Z p rmm × Z s . By Proposition 2.5, Aut( Z n ) = Aut( Z p r ) × Aut( Z p r ) × · · · × Aut( Z p rmm ) × Aut( Z s ). Recall that Hol( Z n ) = R ( Z n )Aut( Z n ). Further-more, R ( x ) α = R ( x α ) for any α ∈ Aut( Z n ) and x ∈ Z n , and ( z · · · z i − z i z i +1 · · · z n ) α i = z · · · z i − z α i i z i +1 · · · z m with z j ∈ Z p rjj for any α i ∈ Aut( Z p rii ). Then [ R ( Z p rjj ) , Aut( Z p rii )] =[ R ( Z s ) , Aut( Z p rii )] = 1 for any i = j , that is, Aut( Z p rii ) commutes with R ( Z p rjj ) and R ( Z s ) elementwise. Since Γ = Cay( Z n , S ) is normal, Proposition 2.1 implies that A = R ( Z n )Aut( Z n , S ) ≤ Hol( Z n ) and A = Aut( Z n , S ).Let G be a regular subgroup of A with G ∼ = Z n . To prove that Γ is a CI-digraph, byProposition2.2 it suffices to show that G = R ( Z n ).Note that G has a unique Sylow p i -subgroup G p i with | G p i | = p r i i , and G ≤ A ≤ Hol( Z n ) = R ( Z n ) ⋊ (Aut( Z p r ) × Aut( Z p r ) × · · · × Aut( Z p rmm ) × Aut( Z s )). Claim 1:
Assume that g = R ( a ) α α · · · α m α m +1 ∈ G , where a ∈ Z n , α m +1 ∈ Aut( Z s )and α i ∈ Aut( Z p rii ) for each 1 ≤ i ≤ m , and assume that R ( Z p rkk ) ≤ G for some1 ≤ k ≤ m . Then α k = 1.Since R ( Z p rkk ) ≤ G and G ∼ = Z n , we have [ g, R ( Z p rkk )] = 1. Since [Aut( Z p rii ) , R ( Z p rkk )] =1 for any i = k and [Aut( Z s ) , R ( Z p rkk )] = 1, we have [ α i , R ( Z p rkk )] = 1 for any i = k ,and since [ R ( a ) , R ( Z p rkk )] = 1 and g = R ( a ) α α · · · α m α m +1 , we have [ α k , R ( Z p rkk )] = 1.This implies that α k = 1 as α k ∈ Aut( Z p rkk ). Claim 2: R ( Z p rkk ) ≤ G for each 1 ≤ k ≤ m . ORMAL CAYLEY DIGRAPHS OF CYCLIC GROUPS WITH CI-PROPERTY 9
Recall that Hol( Z n ) p i is a Sylow p i -subgroup of Hol( Z n ) and R ( Z n ) E Hol( Z n ) = R ( Z n )Aut( Z n ). It follows that Hol( Z n ) p i ≤ R ( Z n )(Aut( Z n )) p i for each 1 ≤ i ≤ m ,and since Aut( Z n ) is abelian, (Aut( Z n )) p i is the unique Sylow p i -subgroup of Aut( Z n ).Since Aut( Z p rii ) ∼ = Z ( p i − p ri − i , Aut( Z p rii ) has a unique subgroup of order p i −
1, de-noted by Aut( Z p rii ) p i − , and a unique Sylow p i -subgroup, that is, Aut( Z p rii ) p i . Thus,Aut( Z p rii ) = Aut( Z p rii ) p i − × Aut( Z p rii ) p i . Since p > p > · · · > p m , we have Aut( Z n ) p i ≤ Aut( Z p r ) p − × Aut( Z p r ) p − × · · · × Aut( Z p ri − i − ) p i − − × Aut( Z p rii ) p i , and in particular,Aut( Z n ) p = Aut( Z p r ) p .We process the proof by induction on k . For k = 1, we argue by contradiction andwe suppose that R ( Z p r ) (cid:2) G . Since | G p | = p r = | R ( Z p r ) | , there exists g ∈ G p but g R ( Z p r ). Note that G p ≤ Hol( Z n ) p , Aut( Z n ) p = Aut( Z p r ) p , and Hol( Z n ) p ≤ R ( Z n )(Aut( Z n )) p = R ( Z n )(Aut( Z p r )) p . Since g ∈ G p , we may write g = R ( g ) α forsome g ∈ Z n and α ∈ (Aut( Z n )) p . Since g is a p -element and R ( Z p r ) is the uniqueSylow p -subgroup of R ( Z n ), g R ( Z p r ) implies that g R ( Z n ). This yields that α = 1. Since α = R ( g ) − g ∈ R ( Z n ) G ≤ Aut(Γ) and α fixes 1 as α ∈ (Aut( Z n )) p , wehave that α ∈ A = Aut( Z n , S ) and hence Aut( Z n , S ) contains an element of order p inAut( Z p r ). By Lemma 3.1, Γ is non-normal, and this is impossible because Γ is normal.Thus, R ( Z p r ) ≤ G , and the claim is true for k = 1.Let k >
1. By inductive hypothesis, we assume that R ( Z p rii ) ≤ G for any 1 ≤ i < k .We suppose that R ( Z p rkk ) (cid:2) G and will obtain a contradiction. Since | G p k | = p r k k ,there exists h ∈ G p k but h R ( Z p rkk ). Since h ∈ Hol( Z n ) p k ≤ R ( Z n )(Aut( Z n )) p k andAut( Z n ) p k ≤ Aut( Z p r ) p − × Aut( Z p r ) p − × · · · × Aut( Z p rk − k − ) p k − − × Aut( Z p rkk ) p k , wehave h = R ( h ) β for some h ∈ Z n and β ∈ (Aut( Z n )) p k , where β = β β · · · β k forsome β i ∈ Aut( Z p rii ) p i − with 1 ≤ i < k and β k ∈ Aut( Z p rkk ) p k . Note that h is a p k -element and R ( Z p rkk ) is the unique Sylow p k -subgroup of R ( Z n ). Then h R ( Z p rkk )implies that h R ( Z n ), forcing β = 1. Since R ( Z p rii ) ≤ G for any 1 ≤ i < k , byClaim 1 we have β i = 1. It follows β = β k = 1, and β k = R ( h ) − h ∈ R ( Z n ) G ≤ A .Furthermore, β k ∈ A = Aut( Z n , S ), and hence Aut( Z n , S ) contains an element of order p k in Aut( Z p rkk ). By Lemma 3.1, Γ is non-normal, a contradiction. Thus, R ( Z p rkk ) ≤ G .By induction, R ( Z p rkk ) ≤ G for each 1 ≤ k ≤ m , as claimed.To prove G = R ( Z n ) = R ( Z p r ) × R ( Z p r ) × · · · × R ( Z p rmm ) × R ( Z s ) with s ≤
2, byClaim 2 we are only left to show R ( Z s ) ≤ G , and it suffices to show that G ≤ R ( Z s ),because | G | = 2 s .Let x ∈ G . Since G ≤ A ≤ Hol( Z n ) = R ( Z n ) ⋊ (Aut( Z p r ) × Aut( Z p r ) × · · · × Aut( Z p rmm ) × Aut( Z s )), we have x = R ( x ) γ γ · · · γ m α , where x ∈ Z n , α ∈ Aut( Z s )and γ i ∈ Aut( Z p rii ) p i − for 1 ≤ i ≤ m . By Claims 2 and 1, γ i = 1 for each 1 ≤ i ≤ m , andhence x = R ( x ) α ∈ R ( Z n )Aut( Z s ). Since [ R ( Z p rii ) , Aut( Z s )] = 1, R ( Z n )Aut( Z s ) hasa unique Sylow 2-subgroup, that is, R ( Z s )Aut( Z s ). It follows that x ∈ R ( Z s )Aut( Z s ),and hence G ≤ R ( Z s )Aut( Z s ).If Aut( Z s ) = 1, then G ≤ R ( Z s ), as required. We may assume that Aut( Z s ) = 1,and hence s = 2 as s ≤
2. It follows that Aut( Z s ) = Aut( Z ) ∼ = Z . In particular, R ( Z s )Aut( Z s ) is a dihedral group of order 8 and hence has a unique cyclic subgroup of order 4, that is, R ( Z s ). Since G ∼ = Z n , G is a cyclic group of order 4, and since G ≤ R ( Z s )Aut( Z s ), we obtain that G = R ( Z s ), as required. This completes theproof. (cid:3) Proof of Corollary 1.3:
Let Z n be the cyclic group of order n . If n = 8 then byProposition 2.4, Z n is a CI-group and hence a NCI-group; if 8 ∤ n then by Theorem 1.2, Z n is a NDCI-group and hence a NCI-group. This proves the sufficiency of the corollary,and the necessity follows from Lemma 3.2. (cid:3) Acknowledgements:
The work was supported by the National Natural ScienceFoundation of China (11731002) and the 111 Project of China (B16002).
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