Two-distance transitive normal Cayley graphs
aa r X i v : . [ m a t h . C O ] F e b TWO-DISTANCE TRANSITIVE NORMAL CAYLEY GRAPHS
JUN-JIE HUANG, YAN-QUAN FENG, JIN-XIN ZHOU
Abstract.
In this paper, we construct an infinite family of normal Cayley graphs,which are 2-distance-transitive but neither distance-transitive nor 2-arc-transitive. Thisanswers a question raised by Chen, Jin and Li in 2019 and corrects a claim in a literaturegiven by Pan, Huang and Liu in 2015. k eywords. Cayley graph, 2-distance-transitive graph, simple group. Introduction
In this paper, all graphs are finite, simple, and undirected. For a graph Γ , let V ( Γ ) , E ( Γ ) , A ( Γ ) or Aut ( Γ ) denote its vertex set, edge set, arc set and its full automor-phism group, respectively. The graph Γ is called G - vertex-transitive , G - edge-transitive or G - arc-transitive , with G ≤ Aut ( Γ ), if G is transitive on V ( Γ ) , E ( Γ ) or A ( Γ ) respec-tively, and G - semisymmetric , if Γ is G -edge-transitive but not G -vertex-transitive. Itis easy to see that a G -semisymmetric graph Γ must be bipartite such that G has twoorbits, namely the two parts of Γ , and the stabilizer G u for any u ∈ V ( Γ ) is transitivethe neighbourhood of u in Γ . An s - arc of Γ is a sequence v , v , . . . , v s of s + 1 vertices of Γ such that v i − , v i are adjacent for 1 ≤ i ≤ s and v i − = v i +1 for 1 ≤ i ≤ s −
1. If Γ hasat least one s -arc and G ≤ Aut ( Γ ) is transitive on the set of s -arcs of Γ , then Γ is called( G, s ) -arc-transitive , and Γ is said to be s -arc-transitive if it is ( Aut ( Γ ) , s )-arc-transitive.For two vertices u and v in V ( Γ ), the distance d ( u, v ) between u and v in Γ isthe smallest length of paths between u and v , and the diameter diam ( Γ ) of Γ is themaximum distance occurring over all pairs of vertices. For i = 1 , , · · · , diam ( Γ ), denoteby Γ i ( u ) the set of vertices at distance i with vertex u in Γ . A graph Γ is called distancetransitive if, for any vertices u, v, x, y with d ( u, v ) = d ( x, y ), there exists g ∈ Aut ( Γ ) suchthat ( u, v ) g = ( x, y ). The graph Γ is called ( G, t ) -distance-transitive with G ≤ Aut ( Γ )if, for each 1 ≤ i ≤ t , the group G is transitive on the ordered pairs of form ( u, v ) with d ( u, v ) = i , and Γ is said to be t -distance-transitive if it is ( Aut ( Γ ) , t )-distance-transitive.Distance-transitive graphs were first defined by Biggs and Smoth in [2], and theyshowed that there are only 12 trivalant distance-transitive graphs. Later, distance-transitive graphs of valencies 3, 4, 5, 6 and 7 were classified in [2, 10, 11, 12, 13], anda complete classification of distance-transitive graphs with symmetric or alternatinggroups of automorphisms was given by Liebeck, Praeger and Saxl [20]. The 2-distance-transitive but not 2-arc-transitive graphs of valency at most 6 were classified in [5,18], and the 2-distance-primitive graphs (a vertex stabilizer of automorphism groupis primitive on both the first step and the second step neighbourhoods of the vertex) with prime valency were classified in [17]. By definition, a 2-arc-transitive graph is2-distance-transitive, but a 2-distance-transitive graph may not be 2-arc-transitive; asimple example is the complete multipartite graph K , . Furthermore, Corr, Jin andSchneider [4] investigated properties of a connected ( G, G, G and a subset S ⊆ G \ { } with S = S − := { s − | s ∈ S } , the Cayley graph
Cay ( G, S ) of the group G with respect to S is the graph with vertex set G and with two vertices g and h adjacent if hg − ∈ S . For g ∈ G , let R ( g ) be thepermutation of G defined by x xg for all x ∈ G . Then R ( G ) := { R ( g ) | g ∈ G } is aregular group of automorphisms of Cay ( G, S ). It is known that a graph Γ is a Cayleygraph of G if and only if Γ has a regular group of automorphisms on the vertex set whichis isomorphic to G ; see [1, Lemma 16.3] and [25]. A Cayley graph Γ = Cay ( G, S ) is called normal if R ( G ) is a normal subgroup of Aut ( Γ ). The study of normal Cayley graphswas initiated by Xu [27] and has been investigated under various additional conditions;see [9, 24].There are many interesting examples of arc-transitive graphs and 2-arc-transitivegraphs constructed as normal Cayley graphs. However, the status for 2-distance-transitivegraphs is different. Recently, 2-distance-transitive circulants were classified in [3], wherethe following question was proposed: Question 1.1. ([3, Question 1.2])
Is there a normal Cayley graph which is 2-distance-transitive, but neither distance-transitive nor 2-arc-transitive?
In this paper, we answer the above question by constructing an infinite family of suchgraphs.
Theorem 1.2.
For an odd prime p , let G = h a, b, c | a p = b p = c p = 1 , [ a, b ] = c, [ c, a ] =[ c, b ] = 1 i and S = { a i , b i | ≤ i ≤ p − } . Then Cay ( G, S ) is a -distance-transitivenormal Cayley graph that is neither distance-transitive nor -arc-transitive. Applying this theorem, we can obtain the following corollary.
Corollary 1.3.
Under the notation given in
Theorem 1.2 , let
Cos ( G, h a i , h b i ) be thegraph with vertex set {h a i g | g ∈ G } ∪ {h b i h | h ∈ G } and with edges all these cosetpairs {h a i g, h b i h } having non-empty intersection in G . Then Cay ( G, S ) is the line graphof Cos ( G, h a i , h b i ) , and Cos ( G, h a i , h b i ) is -arc-transitive. The graph
Cos ( G, h a i , h b i ) was first constructed in [21] as a regular cover of K p,p ,where it is said that Cos ( G, h a i , h b i ) is 2-arc-transitive in [21, Theorem 1.1], but not 3-arc-transitive generally for all odd primes p in a remark after [21, Example 4.1]. However, thisis not true and Corollary 1.3 implies that Cos ( G, h a i , h b i ) is always 3-arc-transitive foreach odd prime p . In fact, Cos ( G, h a i , h b i ) is 3-arc-regular, that is, Aut ( Cos ( G, h a i , h b i ))is regular on the set of 3-arcs of Cos ( G, h a i , h b i ).2. Preliminaries
In this section we list some preliminary results used in this paper. The first one is thewell-known orbit-stabilizer theorem (see [8, Theorem 1.4A]).
WO-DISTANCE TRANSITIVE NORMAL CAYLEY GRAPHS 3
Proposition 2.1.
Let a group G has a transitive action on a set Ω and let α ∈ Ω . Then | G | = | Ω || G α | . The well-known Burnside p a q b theorem was given in [15, Theorem 3.3]. Proposition 2.2.
Let p and q be primes and let a and b be positive integers. Then agroup of order p a q b is soluble. The next proposition is an important property of a non-abelian simple group actingtransitively on a set with cardinality a prime-power, and we refer to [16, Corollary 2] or[26, Proposition 2.4].
Proposition 2.3.
Let T be a nonabelian simple group acting transitively on a set Ω withcardinality a p -power for a prime p . If p does not divide the order of a point-stabilizerof T , then T acts -transitively on Ω . Let Γ = Cay ( G, S ) be a Cayley graph of a group G with respect to S . Then R ( G )is a regular subgroup of Aut ( Γ ), and Aut ( G, S ) := { α ∈ Aut ( G ) | S α = S } is also asubgroup of Aut ( Γ ), which fixes 1. Furthermore, R ( G ) is normalized by Aut ( G, S ), andhence we have a semiproduct R ( G ) ⋊ Aut ( G, S ), where R ( g ) α = R ( g α ) for any g ∈ G and α ∈ Aut ( G, S ). Godsil [14] proved that the semiproduct R ( G ) ⋊ Aut ( G, S ) is in factthe normalizer of R ( G ) in Aut ( Γ ). By Xu [27], we have the following proposition. Proposition 2.4.
Let Γ = Cay ( G, S ) be a Cayley graph of a finite group G with respectto S , and let A = Aut ( Γ ) . Then the following hold: (1) N A ( R ( G )) = R ( G ) ⋊ Aut ( G, S ) ; (2) Γ is a normal Cayley graph if and only if A = Aut ( G, S ) , where A is thestabilizer of in A . Let Γ be a G -vertex-transitive graph, and let N be a normal subgroup of G . The normal quotient graph Γ N of Γ induced by N is defined to be the graph with vertex setthe orbits of N and with two orbits B, C adjacent if some vertex in B is adjacent tosome vertex in C in Γ . Furthermore, Γ is called a normal N -cover of Γ N if Γ and Γ N have the same valency. Proposition 2.5.
Let Γ be a connected G -vertex-transitive graph and let N be a normalsubgroup of G . If Γ is a normal N -cover of Γ N , then the following statements hold: (1) N is semiregular on V Γ and is the kernel of G acting V ( Γ N ) , so G/N ≤ Aut ( Γ N ) ; (2) Γ is ( G, s ) -arc-transitive if and only if Γ N is ( G/N, s ) -arc-transitive; (3) G α ∼ = ( G/N ) δ for any α ∈ V Γ and δ ∈ V ( Γ N ) .In particular, the above results hold if we replace the assumption that Γ is a normal N -cover of Γ N by the following assumption: Γ is G -arc-transitive with a prime valencyand N has at least three orbits. Proposition 2.5 was given in many papers by replacing the condition that Γ is a normal N -cover of Γ N by one of the following assumptions: (1) N has at least 3-orbits and G is2-arc-transitive (see [23, Theorem 4.1]); (2) N has at least 3-orbits, G is arc-transitiveand Γ has a prime valency (see [22, Theorem 2.5]); (3) N has at least 3-orbits and G islocally primitive (see [19, Lemma 2.5]). The first step for these proofs is to show thatfor any two vertices B, C ∈ V ( Γ N ), the induced subgraph [ B ] of B in Γ has no edge andif B and C are adjacent in Γ N then the induced subgraph [ B ∪ C ] in Γ is a matching, HUANG, FENG, AND ZHOU which is equivalent to that Γ is a normal N -cover of Γ N . Then Proposition 2.5 (1)-(3)follows from these proofs. 3. Proof Theorem 1.2
For a positive integer n and a prime p , we use Z n and Z rp to denote the cyclic group oforder n and the elementary abelian group of order p r , respectively. In this section, wealways assume that p is an odd prime, and denote by Z ∗ p the multiplicative group of Z p consisting of all non-zero numbers in Z p . Note that Z ∗ p ∼ = Z p − . Furthermore, we alsoset the following assumptions in this section: G = h a, b, c | a p = b p = c p = 1 , [ a, b ] = c, [ c, a ] = [ c, b ] = 1 i , S = { a i , b i | ≤ i ≤ p − } , Γ = Cay ( G, S ) , A = Aut ( Γ ) , N = N A ( R ( G )) = R ( G ) ⋊ Aut ( G, S ) , and Z ∗ p = h t i . By Proposition 2.4, N A ( R ( G )) = R ( G ) ⋊ Aut ( G, S ), and R ( g ) δ = R ( g δ ) for any R ( g ) ∈ R ( G ) and δ ∈ Aut ( G, S ). Since G = h S i , Γ is a connected Cayley graph ofvalency 2( p − α : a a t , b b, c c t ; β : a a, b b t , c c t ; γ : a b, b a, c c − . Since a t , b, c t satisfy the same relations as a, b, c in G and G = h a t , b, c t i , α induces anautomorphism of G , and we still denote by α this automorphism. Similarly, β and γ arealso automorphisms of G . Lemma 3.1.
Aut ( G, S ) = h α, β, γ i ∼ = ( Z p − × Z p − ) ⋊ Z , and Γ is N -arc-transitive.Furthermore, N has no normal subgroup of order p .Proof. Since Z ∗ p = h t i , it is easy to check that α p − = β p − = γ = 1, αβ = βα and α γ = β . Thus h α, β, γ i ∼ = ( Z p − × Z p − ) ⋊ Z . Clearly, α, β, γ ∈ Aut ( G, S ). To prove
Aut ( G, S ) = h α, β, γ i ∼ = ( Z p − × Z p − ) ⋊Z , it suffices to show that | Aut ( G, S ) | ≤ p − .Clearly, h α, β, γ i is transitive on S , and hence Γ is N -arc-transitive. Since G = h S i , Aut ( G, S ) is faithful on S . By Proposition 2.1, | Aut ( G, S ) | = | S || Aut ( G, S ) a | , where Aut ( G, S ) a is the stabilizer of a in Aut ( G, S ). Note that
Aut ( G, S ) a fixes a i for each1 ≤ i ≤ p −
1. Again by Proposition 2.1, | Aut ( G, S ) a | ≤ ( p − | Aut ( G, S ) a,b | , where Aut ( G, S ) a,b is the subgroup of Aut ( G, S ) fixing a and b . Since G = h a, b i , we obtain Aut ( G, S ) a,b = 1, and then | Aut ( G, S ) | ≤ p − , as required.Let H ≤ N be a subgroup of order p . Since R ( G ) is the unique normal Sylow p -subgroup of N = R ( G ) ⋊ Aut ( G, S ), we have H ≤ R ( G ), and since | R ( G ) : H | = p , wehave H E R ( G ). Note that the center C := Z ( R ( G )) = h R ( c ) i and C ∩ H = 1. Thus, C ∩ H = C as | C | = p , implying C ≤ H . Since H/C is a subgroup of order p , and R ( G ) /C = h R ( a ) C i × h R ( b ) C i ∼ = Z p , we have H/C = h R ( b ) C i or h R ( a ) R ( b ) i C i for some0 ≤ i ≤ p −
1. It follows that H = h R ( b ) i × C or h R ( ab i ) i × C for some 0 ≤ i ≤ p − H E N . Since C is characteristic in R ( G ) and R ( G ) E N , we have C E N .Recall that R ( a ) γ = R ( a γ ) = R ( b ). Then ( h R ( a ) i × C ) γ = h R ( b ) i × C . This impliesthat both h R ( a ) i × C and h R ( b ) i × C are not normal in N . Thus, H = h R ( ab i ) i × C for some 1 ≤ i ≤ p −
1. Since H E N , we have H β = H , that is, h R ( ab ti ) i × C = H β = H = h R ( ab i ) i × C . It follows that h R ( ab ti ) i = h R ( ab i ) i and then R ( ab ti ) = R ( ab i ), which WO-DISTANCE TRANSITIVE NORMAL CAYLEY GRAPHS 5 further implies b ti = b i . This gives rise to p (cid:12)(cid:12) i ( t − i, p ) = 1, we have t = 1,contradicting that Z ∗ p = h t i ∼ = Z p − . Thus, N has no normal subgroup of order p . (cid:3) A clique of a graph Γ is a maximal complete subgraph, and the clique graph Σ of Γ is defined to have the set of all cliques of Γ as its vertex set with two cliques adjacentin Σ if the two cliques have at least one common vertex. For a positive integer n , n p denotes the largest p -power diving n . Lemma 3.2.
The clique graph Σ of Γ is a connected p -valent bipartite graph of order p , A has a faithful natural action on Σ , and Σ is R ( G ) -semisymmetric and N -arc-transitive. Furthermore, | A | p = p .Proof. Recall that G = h a, b, c | a p = b p = c p = 1 , [ a, b ] = c, [ c, a ] = [ c, b ] = 1 i and S = { a i , b i | ≤ i ≤ p − } . Then Γ = Cay ( G, S ) has exactly two cliques passingthrough 1, that is, the induced subgraphs of h a i and h b i in Γ . Since R ( G ) ≤ Aut ( Γ )is transitive on vertex set, each clique of Γ is an induced subgraph of the coset h a i x or h b i x for some x ∈ G . Thus, we may view the vertex set of Σ as {h a i x, h b i x | x ∈ G } with two cosets adjacent in Σ if they have non-empty intersection. It is easy to see that h a i x ∩ h b i y = ∅ if and only if |h a i x ∩ h b i y | = 1, and any two distinct cosets, either in {h a i x | x ∈ G } or in {h b i x | x ∈ G } , have empty intersection. Furthermore, h a i hasnon-empty intersection with exactly p cosets, that is, h b i a i for 0 ≤ i ≤ p −
1. Thus, Σ isa p -valent bipartite graph of order 2 p . The connectedness of Σ follows from that of Γ .Clearly, A has a natural action on Σ . Let K be the kernel of A on Σ . Then K fixeseach coset of h a i x and h b i x for all x ∈ G . Since h a i x ∩ h b i x = { x } , K fixes x and hence K = 1. Thus, A is faithful on Σ and we may let A ≤ Aut ( Σ ).Note that R ( G ) is not transitive on {h a i x, h b i x | x ∈ G } , but transitive on {h a i x | x ∈ G } and {h b i x | x ∈ G } . Furthermore, R ( h a i ) fixes h a i and is transitive on {h b i a i | ≤ i ≤ p − } , the neighbourhood of h a i in Σ , and similarly, R ( h b i ) fixes h b i and is transitiveon the neighbourhood {h a i b i | ≤ i ≤ p − } of h b i in Σ . It follows that Σ is R ( G )-semisymmetric. Recall that N = R ( G ) ⋊ Aut ( G, S ) and
Aut ( G, S ) = h α, β, γ i . Since a γ = b and b γ = a , γ interchanges {h a i x | x ∈ G } and {h b i x | x ∈ G } . This yields that Σ is R ( G ) ⋊ h γ i -arc-transitive and hence N -arc-transitive.Since Σ is a connected graph with prime valency p , we have p ∤ | Aut ( Σ ) u | for any u ∈ V ( Σ ), and in particular, p ∤ | A u | . Note that p (cid:12)(cid:12) | A u | . By Proposition 2.1, | A | = | Σ || A u | = 2 p | A u | . This implies that | A | p = p . (cid:3) Lemma 3.3. A = Aut ( Γ ) = R ( G ) ⋊ Aut ( G, S ) .Proof. By Lemma 3.2, | A | p = p , and since | V ( Γ ) | = p and A is vertex-transitiveon V ( Γ ), the vertex stabilizer A is a p ′ -group, that is, p ∤ | A | . To prove the lemma,by Proposition 2.4 we only need to show that R ( G ) E A , and since R ( G ) is a Sylow p -subgroup of A , it suffices to show that A has a normal Sylow p -subgroup.Let M be a minimal normal subgroup of A . Then M = T × T · · · × T d , where T i ∼ = T for each 1 ≤ i ≤ d with a simple group T . Since | V ( Γ ) | = p , each orbit of M haslength a p -power and hence each orbit of T i has length a p -power. It follows that p (cid:12)(cid:12) | T | .Assume that | T | p = p ℓ . Then | M | p = p dℓ and dℓ = 1 , | A | p = p .We process the proof by considering the two cases: M is insoluble or soluble. Case 1: M is insoluble. HUANG, FENG, AND ZHOU
In this case, T is a non-abelian simple group. We prove that this case cannot happenby deriving contradictions. Recall that dℓ = 1 , dℓ = 1. Then | M | p = p . By Lemma 3.2, M E A ≤ Aut ( Σ ), and since | V ( Σ ) | = 2 p , M has at least three orbits. Since Σ has valency p , Proposition 2.5 impliesthat M is semiregular on V ( Σ ) and hence | M | (cid:12)(cid:12) p . By Proposition 2.2, M is soluble,a contradiction.Assume that dℓ = 2. Since R ( G ) is a Sylow p -subgroup of A and M E A , R ( G ) ∩ M is a Sylow p -subgroup of M and hence | R ( G ) ∩ M | = | M | p = p . Since R ( G ) E N and M E A , M ∩ R ( G ) is a normal subgroup of order p in N , contradicting to Lemma 3.1.Assume that dℓ = 3. Then ( d, ℓ ) = (1 ,
3) or (3 , | M | p = p = | A | p , we deduce R ( G ) ≤ M and hence M is transitive on Γ .For ( d, ℓ ) = (1 , M is a non-abelian simple group. Since M ≤ A is a p ′ -group,Proposition 2.3 implies that M is 2-transitive on Γ , forcing that Γ is the complete graphof order p , a contradiction.For ( d, ℓ ) = (3 , M = T × T × T . Then | M | p = p , and since M E A ,we derive R ( G ) ≤ M . By Lemma 3.2 M ≤ Aut ( Σ ), and Σ is R ( G )-semisymmetric.Since M has no subgroup of index 2, M fixes the two parts of Σ setwise, and hence Σ is M -semisymmetric. Noting that γ interchanges the two parts of Σ , we have that Σ is M h γ i -arc-transitive. Since γ is an involution, under conjugacy it fixes T i for some1 ≤ i ≤
3, say T . Then T E h M, γ i and by Proposition 2.5, T is semiregular on Σ .This gives rise to | T | (cid:12)(cid:12) p , contrary to the simplicity of T . Case 2: M is soluble.Since p (cid:12)(cid:12) | M | , we have M = Z dp with 1 ≤ d ≤
3. If d = 3 then A has a normal Sylow p -subgroup, as required. If d = 2 then M ≤ R ( G ) ≤ N and N has a normal subgroupof order p , contrary to Lemma 3.1. Thus, we may let d = 1, and since M ≤ R ( G ) and R ( G ) has a unique normal subgroup of order p that is the center of R ( G ), we derivethat M = h R ( c ) i .Now it is easy to see that the quotient graph Γ M = Cay ( G/M, S/M ) with
S/M = { a i M, b i M | ≤ i ≤ p − } . Note that G/M = h aM i × h bM i ∼ = Z p . Then Γ M is aconnected Cayley graph of order p with valency 2( p − Γ is a normal M -cover of Γ M . By Proposition 2.5, we may let A /M ≤ Aut ( Γ M ) and Γ M is A /M -arc-transitive.Let H/M be a minimal normal subgroup of A /M . Then H E A and H/M = L /M ×· · · × L r /M , where L i E H and L i /M (1 ≤ i ≤ r ) are isomorphic simple groups. Since | Γ M | = p , we infer p (cid:12)(cid:12) | H/M | and similarly, p (cid:12)(cid:12) | L i /M | . Let | L i /M | p = p s . Then | H/M | p = p rs , and since | A /M | p = p , we obtain that sr = 1 or 2.We finish the proof by considering the two subcases: H/M is insoluble or soluble.
Subcase 2.1:
H/M is insoluble.In this subcase, L i /M are isomorphic non-abelain simple groups. We prove this sub-case cannot happen by deriving contradictions. Recall that sr = 1 or 2.Let sr = 1. Then | H/M | p = p , and therefore | H | p = p . Since H E A , H ∩ R ( G ) isa Sylow p -subgroup of H , implying | H ∩ R ( G ) | = p , and then R ( G ) E N yields that H ∩ R ( G ) is a normal subgroup of order p in N , contrary to Lemma 3.1.Let rs = 2. Then | H/M | p = p and | H | p = p . This yields R ( G ) ≤ H and H istransitive on Γ , so H/M is transitive on V ( Γ M ). Note that ( r, s ) = (1 ,
2) or (2 , WO-DISTANCE TRANSITIVE NORMAL CAYLEY GRAPHS 7
For ( r, s ) = (1 , H/M is a nonabelian simple group. By Propostion 2.5, (
H/M ) u for u ∈ V ( Γ M ) is a p ′ -group because H ≤ A is a p ′ -group, and by Proposition 2.3, H/M is2-transitive on V ( Γ M ), forcing that Γ M is a complete group of order p , a contradiction.For ( r, s ) = (2 , H/M ∼ = L /M × L /M , where L /M and L /M are isomorphicnonabelain simple groups and | L i /M | p = p . It follows that | H | p = p and | L i | p = p for1 ≤ i ≤
2. Since H E A , we derive R ( G ) ≤ H . Note that H has no subgroup of index2. Since Σ is bipartite, it is H -semisymmetric. Let ∆ and ∆ be the two parts of Σ .Then | ∆ | = | ∆ | = p , and H is transitive on both ∆ and ∆ .Suppose ( L ) u = 1 for some u ∈ V ( Σ ) = ∆ ∪ ∆ . By Proposition 2.1, | L | = | u L | ,and since L E H and | ∆ | = | ∆ | = p , we derive | L | = p or p , contrary to theinsolubleness of L . Thus ( L ) u = 1. Since Σ has prime valency p , H u is primitive onthe neighbourhood Σ ( u ) of u in Σ , and since ( L ) u E H u , ( L ) u is transitive on Σ ( u ),which implies that | ( L ) u | p = p . Since | L | p = p , each orbit of L on ∆ or ∆ haslength p .Let x ∈ ∆ and y ∈ ∆ be adjacent in Σ , and let ∆ and ∆ be the orbits of L containing x and y , respectively. Then | ∆ | = | ∆ | = p . Since ( L ) x is transitive on Σ ( x ), x is adjacent to each vertex in ∆ , and therefore, each vertex in ∆ is adjacentto each vertex in ∆ , that is, the induced subgroup [∆ ∪ ∆ ] is the complete bipartitegraph K p,p . It follows that Σ ∼ = p K p,p , contrary to the connectedness of Σ . Subcase 2.2:
H/M is soluble.In this case, | H | = p or p . Recall that H E A . If | H | = p then H ≤ R ( G ) and N has normal subgroup of order p , contradicts Lemma 3.1. Thus, | H | = p and A has anormal Sylow p -subgroup, as required. This completes the proof. (cid:3) Now we are ready to finish the proof.
Proof of Theorem 1.2.
By Lemmas 3.1 and 3.3, Γ is a arc-transitive normal Cayleygraph. In particular, Γ is 1-distance transitive. Since S = { a i , b i | ≤ i ≤ p − } , Γ hasgirth 3, so it is not 2-arc-transitive.Recall that G = h a, b, c | a p = b p = c p = 1 , [ a, b ] = c, [ c, a ] = [ c, b ] = 1 i . Clearly, Γ (1) = S = { a i , b i | ≤ i ≤ p − } , Γ (1) = { b j a i , a j b i | ≤ i, j ≤ p − } . Note that
Aut ( G, S ) = h α, β, γ | α p − = β p − = γ = 1 , α β = α, α γ = β i , where a α = a t , b α = b, c α = c t , a β = a, b β = b t , c β = c t , a γ = b, b γ = a and c γ = c − . Then( ba ) α i β j = b t i a t j , and since Z ∗ p = h t i , we obtain that h α, β i is transitive on the set { b j a i | ≤ i, j ≤ p − } . Similarly, h α, β i is transitive on { a j b i | ≤ i, j ≤ p − } .Furthermore, γ interchanges the two sets { b j a i | ≤ i, j ≤ p − } and { a j b i | ≤ i, j ≤ p − } . It follows that Aut ( G, S ) is transitive on Γ (1) and hence Γ is 2-distancetransitive.Noting that ab = bac , we have that b − ab = ac ∈ Γ (1) and aba = ba c ∈ Γ (1). Alsoit is easy to see that ( ac ) Aut ( G,S ) = ( ac ) h α,β,γ i = { a i c j , b i c j | ≤ i, j ≤ p − } . Now it iseasy to see that ba c ( ac ) Aut ( G,S ) , and since A = Aut ( G, S ) by Proposition 2.4, Γ isnot distance-transitive. (cid:3) Proof of Corollary 1.3:
Recall that Σ is the clique graph of Γ . By the first paragraphin the proof of Lemma 3.2 and the definition of Cos ( G, h a i , h b i ) in Corollary 1.3, we HUANG, FENG, AND ZHOU have Σ = Cos ( G, h a i , h b i ). Again by Lemma 3.2, Σ is R ( G )-semisymmetric, and since | E ( Σ ) | = (2 p · p ) / p = | R ( G ) | , R ( G ) is regular on the edge set E ( Σ ) of Σ . Thus,the line graph of Σ is a Cayley graph on G .For a given edge {h a i x, h b i y } ∈ E ( Σ ), we have |h a i x ∩ h b i y | = 1, and then we mayidentify this edge with the unique element in h a i x ∩ h b i y . Note that Σ has valency2( p − h a i ∩ h b i in Σ is exactly incident to all edges in S = { a i , b i | ≤ i ≤ p − } , because { a i } = h a i ∩ h b i a i and { b i } = h b i ∩ h a i b i . It follows that Γ = Cay ( G, S ) is exactly the line graph of Σ .If α ∈ Aut ( Σ ) fixes each edge in Σ then α fixes all vertices of Σ , that is, Aut ( Σ ) actsfaithfully on Γ . Thus, we may view Aut ( Σ ) as a subgroup of Aut ( Γ ). By Lemmas 3.2and 3.3, we have Aut ( Γ ) = Aut ( Σ ) = R ( G ) ⋊ Aut ( G, S ).Recall that
Aut ( G, S ) = h α, β, γ i and Σ is arc-transitive. Since a β = a , b β = b t and c β = c t , where Z ∗ p = h t i , h β i fixes the arc ( h a i , h b i ) in Σ and is transitive on the vertexset {h a i b i | ≤ i ≤ p − } , where {h a i} ∪ {h a i b i | ≤ i ≤ p − } is the neighbourhoodof h b i in Σ . Thus, Σ is 2-arc-transitive. Since a α = a t , b α = b and c α = c t , h α i fixesthe 2-arc ( h a i , h b i , h a i b ) and is transitive on the vertex set {h b i a i b | ≤ i ≤ p − } ,where {h b i} ∪ {h b i a i b | ≤ i ≤ p − } is the neighbourhood of h a i b in Σ . It followsthat Σ is 3-arc-transitive. It is easy to see that the number of 3-arcs in Σ equals to | A | = 2 p ( p − , A is regular on the set of 3-arcs of Σ . (cid:3) References [1] Biggs, N.: Algebraic graph theory, Cambridge Mathematical Library, 2nd edition, CambridgeUniversity Press, Cambridge, 1993[2] Biggs, N., Smith, D.: On trivalent graphs.
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