aa r X i v : . [ m a t h . C O ] J a n ARC-TRANSITIVE CIRCULANTS
SHU JIAO SONG
Abstract.
This short paper presents characterisations of normal arc-transitivecirculants and arc-transitive normal circulants, that is, for a connected arc-transitivecirculant Γ = Cay( C, S ), it is shown that • Aut(
C, S ) is transitive on S if and only if each element of S has order n ; • Aut Γ ✄ C if and only if S does not contain a coset of any subgroup.This completes the classification of arc-transitive circulants given by Li-Xia-Zhou. Introduction
A digraph is an ordered pair (
V, A ) with the vertex set V and arc set A , in whichan arc is an ordered pair of adjacent vertices. The number | V | of vertices is calledthe order of Γ. A digraph Γ is said to be arc-transitive if the automorphism groupAut Γ acts transitively on the arc set.Let C = Z n = { , , . . . , n − } be a cyclic group of order n , and let S be a subsetof C \{ } . Let Cay( C, S ) be the Cayley digraph with vertex set C such that a vertex v is adjacent to a vertex w if and only if w − v ∈ S . In this paper, we sometimesdenote Cay( C, S ) by
Circ ( n, S ) if we do not need to emphasise on the connectingsubset S . A digraph is called a circulant of order n if it is isomorphic to a digraph Circ ( n, S ) for some subset S of Z n .The cyclic group C acts regularly on the vertex set by right multiplication as anautomorphism group. Thus a circulant is a vertex transitive graph. A circulant Γ iscalled a normal circulant if Aut Γ has a cyclic subgroup which is normal and regularon the vertices.Much effort has been made to characterize arc-transitive circulants in the litera-ture: Chao and Wells [3, 4] classified those of prime order; 2-arc-transitive [1, 18],square-free order [15], odd prime-power order [22]. In the general case, a recursivecharacterization of arc-transitive circulants was given by Kov´acs [10] and Li [14],independently. Recently, Li-Xia-Zhou [17] prove that an arc-transitive circulant Γ has the form Γ ∼ = ( Γ × K n × · · · × K n r )[ K b ] , where Γ is a normal circulant of order n and n , n , . . . , n r are pairwise coprime.This would provide a complete characterization of arc-transitive circulants if normalcirculants Γ is well-characterised.For a Cayley graph Γ = Cay( C, S ), there is a useful subgroup of automorphismsAut(
C, S ) = h σ ∈ Aut( C ) | S σ = S i . This was first introduced by Godsil [8]. Obviously, Aut Γ > C :Aut( C, S ). Thismotivated Xu [21] to introduce normal Cayley graphs , which are those satisfying
Aut Γ = C :Aut( C, S ). This was extended by Praeger [19] to define normal arc-transitive Cayley graphs , which are those such that Aut(
C, S ) is transitive on S . Wenotice that a normal arc-transitive Cayley graph is not necessarily a normal Cayleygraph. Praeger [19] proposed to characterize normal arc-transitive Cayley graphs.Obviously, if Aut( C, S ) is transitive on S , then elements of S all have the sameorder. The first main result of this paper shows that the converse statement is alsotrue, which is a bit surprising. Theorem 1.1.
Let Γ = Circ ( n, S ) be a connected arc-transitive circulant. Then Γis normal-arc-transitive if and only if each element of S is of order n . This solves a problem of [19] for cyclic groups. Another version of Theorem 1.1would be interesting.
Corollary 1.2.
Let Γ = Circ ( n, S ) be such that each element of S has order n .Then Γ is arc-transitive if and only if Γ is normal-arc-transitive. In the terminology of [17], normal arc-transitive circulants are characterized below,where π ( n ) for an integer n denotes the set of all prime divisors of n . Corollary 1.3.
A normal arc-transitive circulant has the formΓ = ( Γ × K p × · · · × K p r )[ K b ] , such that p , . . . , p r are distinct primes, and π ( b ) ⊂ π ( n ) ∪ { p , . . . , p r } . By definition, if an arc-transitive circulant is a normal circulant, then it is normalarc-transitive. The next theorem provides a simple criterion for an arc-transitivecirculant to be a normal circulant.
Theorem 1.4.
Let Γ = Circ ( n, S ) be a connected arc-transitive circulant. Then Γis a normal circulant if and only if S does not contain a coset of any subgroup. Corollary 1.5.
An arc-transitive circulant
Circ ( n, S ) of valency coprime to n is anormal circulant. In particular, Circ ( p e , S ) with p e > is normal if and only if | S | divides p − . Corollary 1.6.
Let n = p e . . . p e r r be the prime factorisation. If Circ ( n, S ) is anormal circulant, then the valency | S | divides ( p − p − . . . ( p r − . Normal arc-transitive circulants
Let n be a positive integer, and let φ ( n ) the Euler φ -function, which is the numberof positive integers that are less than and coprime to n . Let C be a cyclic group oforder n . Then the automorphism group Aut( C ) is an abelian group of order φ ( n ). Lemma 2.1. ([8, 21])
For a circulant Γ = Cay(
C, S ) , the normalizer of C in Aut
Γis C ⋊ Aut(
C, S ) . A circulant Γ is called normal-arc-transitive if the normaliser of a cyclic regularsubgroup in Aut Γ is arc-transitive on Γ. Lemma 2.2.
If Γ = Σ [ K b ] is a normal-edge-transitive circulant, then so is thequotient Σ . We note that not every arc-transitive circulant is normal-arc-transitive, for exam-ple, K p ; not every normal-arc-transitive circulant is a normal circulant, for instance, K p × K q where p, q are distinct primes. RC-TRANSITIVE CIRCULANTS 3
Lemma 2.3.
A circulant
Circ ( n, S ) is normal arc-transitive if and only if h S i = Z n and Aut( Z n , S ) is regular on S .Proof. Assume that
Circ ( n, S ) is normal arc-transitive. Then Aut( Z n , S ) is tran-sitive on S , and h S i = Z n . Thus every element of S is a generator of Z n . If σ ∈ Aut( Z n , S ) fixes some element g ∈ S , then since h g i = Z n , σ fixes every elementof Z n , and so σ =. Therefore, Aut( Z n , S ) is regular on S .Conversely, if h S i = Z n and Aut( Z n , S ) is regular on S , then clearly Circ ( n, S )is a normal arc-transitive circulant. ✷ In other words, normal arc-transitive circulants are exactly normal arc-regularcirculants.For two graphs Γ = ( U, E ) and Σ = ( V, F ), the direct product Γ × Σ is the graphwith vertex set U × V such that two vertices ( u , v ) , ( u , v ) are adjacent if andonly if either u , u are adjacent in Γ and v = v , or v , v are adjacent in Σ and u = u . Lemma 2.4.
Let Γ = Cay( G , S ) × Cay( G , S ) . Then Γ = Cay( G × G , T ) ,where T = { ( s , s ) | s i ∈ S i } , and furthermore, the following hold. (i) Γ is a circulant if and only if G , G are cyclic groups of coprime orders. (ii) Elements of T have the same order if and only if elements of S i are all ofthe same order, where i = 1 or .Proof. The proof follows from the definitions of Cayley graphs and direct productof graphs. ✷ Obviously, a Cayley graph Cay(
G, S ) is a complete graph if and only if S = G \{ } .For digraphs Γ = ( V , A ) and Σ = ( V , A ), the lexicographic product Γ [ Σ ] of Σ by Γ is the digraph with vertex set V × V such that ( u , u ) is connected to ( v , v )if and only if either ( u , v ) ∈ A , or u = v and ( u , v ) ∈ A . Lemma 2.5.
Let C = Z n be a cyclic group of order n , and let Γ = Cay( C, S ) beconnected and arc-transitive. Then the following statements are equivalent: (i) Γ = Σ [ K b ] for some graph Σ and some integer b > ; (ii) S = T h g i , where g ∈ C and T ⊂ S , such that | S | = | T || g | with | g | = b .Proof. Assume first that Γ = Σ [ K b ]. Then Σ = Cay( C, S ) is a Cayley graph ofa cyclic group C , where C = C/N with | N | = b , and | S | = | S | b . It follows that S = N s ∪ N s ∪ · · · ∪ N s r such that S = { s , s , . . . , s r } . Let N = h g i . Then S = N { s , s , . . . , s r } = h g i T , where T = { s , s , . . . , s r } .Now assume that S = T h g i , where T ⊂ S . Then C = h g i ∪ g h g i ∪ · · · ∪ g b − h g i , where | g | = b . Let B i = h g i g i , where 0 i b −
1. Suppose g i ∈ S for some integer i . Then g i h g i ⊂ S , and so the vertex (1 ,
1) is adjacent to all vertices in B i = g i h g i .It follows that the induced subgraph [ B , B i ] = K b,b , and so Γ = Σ [ K b ], where Σ isthe quotient graph Γ induced by the block system { B , B , . . . , B b − } . ✷ We now quote the main result of [17].
Theorem 2.6. ([17])
Let Γ be a connected arc-transitive circulant of order n . Then SONG (1) Γ ∼ = ( Γ × K n × · · · × K n r )[ K b ] , and (2) Aut( Γ ) ∼ = Sym( b ) ≀ (Aut( Γ ) × Sym( n ) × · · · × Sym( n r )) ,where Γ ≇ C is a connected normal circulant of order n , n i > for i = 1 , . . . , r ,and n = n n · · · n r b , and n , n , . . . , n r are pairwise coprime. Lemma 2.7.
Let Γ = Circ ( n, S ) be connected and arc-transitive. If S contains acoset of some subgroup, then Γ is not a normal circulant.Proof. Suppose that Γ = Cay( C, S ) is a normal circulant, and t h h i ⊂ S for somenon-identity element h ∈ C Then Aut(
C, S ) is transitive on S , and hence everyelement of S is contained in a coset of h h i . It follows that S = t h h i ∪ · · · ∪ t r h h i .By Lemma 2.5, the graph Γ = Σ [ K b ], and Aut Γ = Sym( b ) ≀ Aut Σ by Theorem 2.6.So b C is not normal in Aut Γ . ✷ Lemma 2.8.
Let Γ = Circ ( n, S ) be such that all elements of S have order n . ThenΓ = ( Γ × K n × · · · × K n r )[ K b ] satisfying Theorem . such that (i) n , . . . , n r are all primes, (ii) π ( b ) ⊂ π ( n ) ∪ { n , . . . , n r } .Proof. By Lemma 2.5, we have S = T h h i where | h | = b . Let T = { t , . . . , t r } besuch that S is a disjoint union of cosets of h h i with representatives in T : S = t h h i ∪ · · · ∪ t r h h i . In particular, t , . . . , t r ∈ S since h h i contains the identity.Since elements of S have the same order n , the elements t , . . . , t r are of orderequal to n . Suppose that a prime divisor p of b does not divide n/b . If | t i | is coprimeto p , then t i h has order divisible by p , which is a contradiction since | t i | = | t i h | .Thus t i = xy where | x | is a power of p and gcd( | x | , | y | ) = 1. Then x ∈ h h i , and so y = t i x − ∈ S , which is not possible since | t i | = | t i x − | . So each prime divisor of b divides n/b , namely, π ( b ) ⊂ π ( n/b ) = π ( n ) ∪ { n , . . . , n r } . ✷ Let C = Z n , and let n = p d . . . p d t t be the prime factorization of n . Let C i be theSylow p i -subgroup of C . Then C i = Z p dii , andAut( C i ) = (cid:26) Z p di − i ( p i − = Z p di − i × Z p i − , if p i is odd , Z × Z di − , if p i = 2.In particular, the Hall p ′ i -subgroup of Aut( C i ) is isomorphic to Z p i − , that is, byAut( C i ) p ′ i = Z p i − . Moreover, Aut( C ) p ′ × · · · × Aut( C t ) p ′ t induces a faithful actionon the factor group C/N . Lemma 2.9.
Let N Φ( C ) , and let C = C/N . Then each automorphism of C isinduced by an automorphism of C , and Aut( C ) is surjective to Aut( C ) .Proof. It is easy to see that Aut( C ) is regular on the set of generators of C = h g i .Since N is a characteristic subgroup of C , the automorphism group Aut( C ) fixes N . Now Aut( C ) acts regularly on the set of generators of C/N = h g i = Z m . Since N Φ( C ), we have π ( m ) = π ( n ). Let g k and g ℓ be two generators of C , andlet σ ∈ Aut( C ) such that ( g k ) σ = g ℓ . Then gcd( k, m ) = gcd( ℓ, m ) = 1, and sogcd( k, n ) = gcd( ℓ, n ) = 1. Thus g k and g ℓ are generators of C . Let σ ∈ Aut( C ) be RC-TRANSITIVE CIRCULANTS 5 such that ( g k ) σ = g ℓ . Then σ induces an automorphism of C which sends g k to g ℓ .Clearly, this is the automorphism σ defined above. ✷ Lemma 2.10.
Let S = g h h i such that all elements of S have order n and h ∈ N where N Φ( C ) . Then Aut(
C, S ) = h τ i is transitive on S , where τ = ( σ , σ , . . . , σ t ) is defined below.Proof. Let n = p d p d . . . p d t t , and let g = g g . . . g t where | g i | = p d i i . Then h = h h . . . h t such that h i = g e i i , where e i is a positiveinteger, and | h i | = p d i − e i i . Let S i = g i h h i i , for 1 i t . Then S i = g i h h i i = { g jp eii i | j < d i − e i } = { g i , g p eii , . . . , g p eii } . Let σ i ∈ Aut( C p i ) be such that g σ i i = g p eii i . Then g h σ i i i = { g (1+ p eii ) k i | k < d i − e i } . Suppose that g (1+ p i ) k i = g (1+ p i ) k i with k > k . Then g (1+ p i ) k − (1+ p i ) k i = 1, and so(1 + p i ) k ((1 + p i ) k − k −
1) = 0 ( mod p d i i ) . So (1 + p i ) k − k ≡ p d i i ). However, as k − k < d i , this is not possible. Thusthe cardinality of { g (1+ p eii ) k i | k < d i − e i } is equal to p e i i , and Aut( h g i i , S i ) = h σ i i is transitive on S i . Since S = { ( s , s , . . . , s t ) | s i ∈ S i } and ( σ , σ , . . . , σ t ) ∈ Aut( C ), it follows that Aut( C, S ) is transitive on S . ✷ Lemma 2.11.
An arc-transitive circulant
Cay(
C, S ) is a normal circulant if andonly if Aut( P i ) p i ∩ Aut(
C, S ) = 1 for i t and Aut( P i ) p ′ i Aut(
C, S ) if P i = Z p i .Proof. Suppose that Aut( P i ) p i ∩ Aut(
C, S ) = h s i 6 = 1. Then for each g ∈ S , wehave g h σ i = g h h i where | h | = | σ | . So S = T h h i , and Γ = ( Γ × K n ×· · · × K n r )[ K b ]with b = | h | , satisfying the conditions given in Theorem 2.6. Thus Γ is not a normalcirculant by Theorem 2.6. If P i = Z p i and Aut( P i ) p ′ i Aut(
C, S ), then Γ = Σ × K p i ,which implies that Γ is not a normal circulant.If Aut( P i ) p i ∩ Aut(
C, S ) = 1 for 1 i t and Aut( P i ) p ′ i Aut(
C, S ) whenever P i = Z p i , then by Theorem 2.6 we have Γ = Γ , which is a normal circulant. ✷ Lemma 2.12.
Let Γ = Circ ( C, S ) be such that all elements of S have order equalto n . Then Aut(
C, S ) is transitive on S , and Γ is a normal arc-transitive circulant.Proof. Since elements of S are all of order n , by Lemma 2.8, Γ = ( Γ × K n ×· · · × K n r )[ K b ] satisfying Theorem 2 . n , . . . , n r are all distinct primes,and π ( b ) ⊂ π ( n ) ∪ { n , . . . , n r } . By Lemma 2.5, the connecting set S has the form S = T h h i , where | h | = b . Let p i = n i for 1 i r , and let π ( n ) = { p r +1 , . . . , p r + s } if n = 1.Let Γ = Cay( n , T ), and K p i = Cay( p i , T i ), where T i = Z p i \ { } . Then T = { ( t , t , . . . , t r ) | t i ∈ T i for 0 i r } . SONG
Thus S = T h h i = ∪ ( t ,t ,...,t r ) ∈ T ( t , t , . . . , t r ) h h i . By Theorem 2.6, the automorphism groupAut( Γ ) ∼ = Sym( b ) ≀ (Aut( Γ ) × Sym( p ) × · · · × Sym( p r ))where p , . . . , p r are distinct primes. For 1 i r + s , let b p i be the p i -part of b . Then h h p i i is a Sylow p i -subgroup of h h i . Since p i divides n/b , there existsan automorphism σ i of a Sylow p i -subgroup C p i of C of order equal to | h p i | . For1 i r , let τ i be an automorphism of C p i of order p i − Γ is a normal circulant, Γ = Cay( C , S ) with C = Z n such thatAut( C , S ) is transitive on S . As π ( b ) ⊂ π ( n ) ∪ { p , . . . , p r } , each prime p i divides n for r + 1 i r + s . Let C π ( n ) be the Hall π ( n )-subgroup of C . Thenby Lemma ?? there exists a subgroup K Aut( C p i ) such that Aut( C , S ) is thequotient of K and K ∼ = Aut( C , S ).Now Γ B = Γ × K p × · · · × K p r = Cay( C, S ), and
X/X ( B ) is arc-transitive on Γ B . Clearly, all elements of S have the same order, and by induction, Aut( C, S ) istransitive on S = { g , . . . , g r } . For any j ∈ { , . . . , r } , there exists σ j ∈ Aut(
C, S )such that g σ j = g j . By Lemma 2.9, there exists σ j ∈ Aut( C ) which is a preimage of σ j . Then ( g h h i ) σ j = g j h h i , and so g σ j = g j h ξ j for some integer ξ j . Let τ be the automorphism defined above, and let H = h τ, σ , . . . , σ r i . Then H Aut(
C, S ) and H is transitive on S . So Γ is a normal arc-transitivecirculant. ✷ Proof of Theorem 1.1:
Assume that Γ = Cay( C, S ) is a normal edge-transitive circulant, where C = Z n .Then Aut( C, S ) is transitive on S , and hence all elements of S have the same order.On the other hand, by Theorem 2.6, Γ ∼ = ( Γ × K n ×· · ·× K n r )[ K b ]. By Lemma 2.2,the quotient Σ = Cay( C, S ) is a norma-edge-transitive circulant. If n i is not a primefor i >
1, then S contains elements of different orders, not possible. Thus each n i isa prime, as in part (ii).Suppose part (ii) holds, namely, Γ ∼ = ( Γ × K p × · · · × K p r )[ K b ], where n = | Γ | and either b = 1 or π ( b ) ⊂ π ( n p . . . p r ) such that p , . . . , p r are distinct primes, n = n p . . . p r b , and gcd( n , p . . . p r ) = 1. Further, if Γ = C , then Aut( Γ ) ∼ =S b ≀ (Aut( Γ ) × S p × · · · × S p r ).3. Normal circulants
Let Γ = Cay( C, S ) be connected and arc-transitive, where C is cyclic.Assume that Γ is a normal circulant, namely, Aut Γ = C :Aut( C, S ). ThenAut(
C, S ) is transitive on S . By Lemma 2.7, S does not contain any coset g h h i .Suppose that S contains g ( h h i ). Then S = { g , . . . , g r }h h i = g h h i ∪ · · · ∪ g r h h i RC-TRANSITIVE CIRCULANTS 7 since Aut(
C, S ) is transitive on S and h h i is a characteristic subgroup of C .Let C = h g i = Z n and n = p e . . . p e r r be a prime decomposition. Let C i be theSylow p i -subgroup of C . Then Aut( C ) = Aut( C ) × · · · × Aut( C r ).Assume that p < p < · · · < p t , and P i is the Sylow p i -subgroup of C . Then C = P × P × · · · × P t with | P i | = p e i i , andAut( C ) = Aut( P ) × Aut( P ) × · · · × Aut( P t ) , and Aut( P i ) = Z ( p i − p ei − i if p i is odd, and Aut( Z e ) = Z × Z e − for e > Lemma 3.1.
An arc-transitive circulant
Cay(
C, S ) is a normal circulant if and onlyif Aut(
C, S ) Z p − × Z p − × · · · × Z p t − .Proof. Assume that Γ = Cay( C, S ) is a normal Cayley graph which is arc-transitive.Then Cay(
C, S ) is an arc-transitive normal circulant, and by Theorem ?? , Γ = ( Γ × K p × · · · × K p r )[ K b ]with n = | Γ | such that either b = 1 or π ( b ) ⊂ π ( n p . . . p r ), where p , . . . , p r aredistinct primes, n = n p . . . p r b , and gcd( n , p . . . p r ) = 1. Proof of Corollary 1.5:
Let
Circ ( p e , S ) be a normal circulant and arc-transitive.Then it is neither a complete graph nor a lexicographic product of a circulant andan isolated graph. Thus gcd( | S | , p ) = 1. Since the automorphism group of a cyclicgroup Z p e is a cyclic group of order p e ( p − | S | divides p − Circ ( p e , S ) is arc-transitive and | S | divides p − p e −
1. Let Γ = Circ ( p e , S ), and let G = Aut Γ .Since | S | divides p −
1, we have Γ = Σ [ K b ]. Since | S | < p e − Γ is not acomplete graph. Since | G | = p e , it follows that Γ = Γ × · · · × Γ m , where m > Circ ( p e , S ) = Γ is a normal circulant. ✷ Let Γ = Circ ( n, S ). If b = 1, then S = S ′ B, where S ′ is subset of S and B is a subgroup of R = Z n . If b = 1 and Γ = Γ × K p with p prime, then S = T Z p , where T is the projection of S in R p ′ . Lemma 3.2.
A circulant is unitary if and only if it is a direct product of its unitarySylow-circulants.Proof.
Let Γ =
Circ ( n, S ) be a unitary circulant of order n . Then Γ has a directproduct decomposition: Circ ( n, S ) = Circ ( p e , S ) × · · · × Circ ( p e r r , S r ) , where each Circ ( p e i i , S i ) is a unitary circulant. ✷ A circulant is normal arc-transitive if and only if all of its Sylow projections arenormal arc-transitive.The unit circulant
Circ ( n, S ) is such that S consists of all generators of Z n . Corollary 3.3.
For n = p e . . . p e r r , the unit circulant of order n is isomorphic to ( K p × · · · × K p r )[ K p e − ...p er − r ] = K p [ K p e − ] × · · · × K p r [ K p er − r ] . SONG
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