Pentavalent symmetric graphs of order twice a prime power
aa r X i v : . [ m a t h . C O ] M a r Pentavalent symmetric graphs of order twice a primepower
Yan-Quan Feng, Jin-Xin Zhou
Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China
Yan-Tao Li
College of Arts and Science, Beijing Union University, Beijing 100191, P.R. China
Abstract
A connected symmetric graph of prime valency is basic if its automorphism groupcontains no nontrivial normal subgroup having more than two orbits. Let p be aprime and n a positive integer. In this paper, we investigate properties of connectedpentavalent symmetric graphs of order 2 p n , and it is shown that a connected pen-tavalent symmetric graph of order 2 p n is basic if and only if it is either a graph oforder 6, 16, 250, or a graph of three infinite families of Cayley graphs on generalizeddihedral groups – one family has order 2 p with p = 5 or 5 | ( p − p with 5 | ( p ± p . Furthermore, theautomorphism groups of these basic graphs are computed. Similar works on cubicand tetravalent symmetric graphs of order 2 p n have been done.It is shown that basic graphs of connected pentavalent symmetric graphs oforder 2 p n are symmetric elementary abelian covers of the dipole Dip , and withcovering techniques, uniqueness and automorphism groups of these basic graphs aredetermined. Moreover, symmetric Z np -covers of the dipole Dip are classified. As abyproduct, connected pentavalent symmetric graphs of order 2 p are classified. Key Words:
Symmetric graph, Cayley graph, regular covering, normal cover.
Let G be a permutation group on a set Ω and α ∈ Ω. Denote by G α the stabilizer of α in G , that is, the subgroup of G fixing the point α . We say that G is semiregular on Ωif G α = 1 for every α ∈ Ω and regular if G is transitive and semiregular. We will use thesymbol Z n , both for the cyclic group of order n and for the ring of integers modulo n (andfor the field of order n if n is a prime). Denote by Z ∗ n the multiplicative group of units of Z n , by D n the dihedral group of order 2 n , and by A n and S n the alternating group andthe symmetric group of degree n , respectively. E-mail addresses: [email protected], [email protected], [email protected] V (Γ), E (Γ) and Aut(Γ) denote the vertex set, edge set and full automorphismgroup of Γ, respectively. An s -arc in a graph Γ is an ordered ( s + 1)-tuple ( v , v , · · · , v s )of s + 1 vertices such that { v i − , v i } ∈ E (Γ) for 1 ≤ i ≤ s and v i − = v i +1 for 1 ≤ i ≤ s −
1, and a 1-arc is also called an arc . For a subgroup G of Aut(Γ) of a graphΓ, the graph Γ is said to be ( G, s )- arc-transitive or ( G, s )- regular if G acts transitivelyor regularly on the set of s -arcs of Γ, and ( G, s )- transitive if G acts transitively on theset of s -arcs but not on the set of ( s + 1)-arcs of Γ. A graph Γ is said to be s - arc-transitive , s - regular or s - transitive if it is (Aut(Γ) , s )-arc-transitive, (Aut(Γ) , s )-regular or(Aut(Γ) , s )-transitive. In particular, 0-arc-transitive means vertex-transitive , and 1-arc-transitive means arc-transitive or symmetric .Let Γ be a graph and N ≤ Aut(Γ). The quotient graph Γ N of Γ relative to N is definedas the graph with vertices the orbits of N on V (Γ) and with two orbits adjacent if thereis an edge in Γ between those two orbits. The theory of quotient graph is widely used toinvestigate symmetric graphs. Let Γ be a symmetric graph and N ⊳ Aut(Γ). If Γ andΓ N have same valency, the graph Γ is said to be a normal cover of Γ N and the graph Γ N is said to be a normal quotient of Γ. In this case, N is semiregular on V (Γ). There aretwo steps to study a symmetric graph Γ — the first step is to investigate normal quotientgraph Γ N for some normal subgroup N of Aut(Γ) and the second step is to reconstructthe original graph Γ from the normal quotient Γ N by using covering techniques. This isusually done by taking the normal subgroup N as large as possible and then the graph Γis reduced to a ‘basic graph’. The situation seems to be somewhat more promising with2-arc-transitive graphs, and the strategy for the structural analysis of these graphs, basedon taking normal quotients, was first laid out by Praeger (see [45, 46, 47]). The strategyworks for locally primitive graphs, that is, vertex-transitive graphs with vertex stabilizersacting primitively on the corresponding neighbors sets (see [48, 49]).As for the first step, let us define some notations. A graph Γ is called basic if Γhas no proper normal quotient. Then a locally primitive graph is basic if and only if ithas no nontrivial normal subgroup having more than two orbits. A graph is quasiprim-itive if every nontrivial normal subgroup of its automorphism group is transitive, and is biquasiprimitive if it has a nontrivial normal subgroup with two orbits but no such sub-group with more than two orbits. Therefore for locally primitive graphs, basic graphs areequivalent to quasiprimitive or biquasiprimitive graphs, which have received most of theattention thus far. In [27], Ivanov and Praeger completed the classification of quasiprim-itive 2-arc-transitive graphs of affine type, and Baddeley gave a detailed description ofquasiprimitive 2-arc-transitive graphs of twisted wreath type [2]. A similar descriptionof 2-arc-transitive graphs associated with Suzuki groups and Ree groups was obtained byFang and Praeger [12, 13]. Classifications of quasiprimitive 2-arc-transitive graphs of oddorder and prime power order have been completed by Li [29, 30, 31], and based on thisapproach, finite vertex-primitive 2-arc-regular graphs have been classified [11] and finite2-arc-transitive Cayley graphs of abelian groups have been determined [33]. Most recently,symmetric graphs of diameter 2 admitting an affine-type quasiprimitive group were inves-tigated by Amarra et al. [1], and an infinite family of biquasiprimitive 2-arc-transitivecubic graphs were constructed by Devillers et al. [7].2ased on the stabilizers of pentavalent symmetric graphs given by Guo and Feng [23],in this paper we prove that normal quotient graphs of connected pentavalent symmetricgraph of order twice a prime power can be K , F Q (the folded hypercube of order 16 ), CD p ( p = 5 or 5 | ( p − CGD p ( p = 5 or 5 | ( p − CGD p (5 | ( p ± CGD p ( p = 5 or5 | ( p − CGD p , where the graphs are defined in Eqs (1)-(6). Automorphism groupsof these normal quotients are computed, and among them, basic ones are determined,which are K , F Q , CGD , CD p ( p = 5 or 5 | ( p − CGD p (5 | ( p ± CGD p ( p = 3 or p ≥ Z p , p a prime, and whosefibre-preserving subgroup of automorphisms acts 2-arc-transitively, were classified. Thisresult has been extended to the case where the group of covering transformations is iso-morphic to Z p , p a prime [8]. Some general methods of elementary abelian coverings weredeveloped in [9, 37, 38]. By using the method developed in [38], Malniˇc and Potoˇcnik [40]classified all vertex-transitive elementary abelian covers of the Petersen graph. Symmet-ric cyclic or elementary abelian covers of the complete graph K , the complete bipartitegraph K , , the cube Q and the Petersen graph O , were classified in [14, 15, 16, 18, 19].Symmetric elementary abelian covers of the unique connected cubic symmetric graph oforder 14, 16 or 18 were classified in [42, 43, 44]. By using the above covers, together withgroup theory techniques, many classifications of symmetric graphs have been obtained –for example, symmetric cubic graphs of order rp or tp were classified for each 2 ≤ r ≤ ≤ t ≤
10. Classification of symmetric graphs with a given order has been widelyinvestigated, and for more results, see [3, 4, 32, 50, 51, 53]. In the above papers, graphsand their covers are simple, that is, no loops and multiple edges. Regular covers of non-simple graphs were also considered in literature and in this case, automorphism groupsof non-simple graphs are usually considered as permutation groups on the sets of arcs ofthese graphs. For example, to classify tetravalent non-Cayley graph of order four times aprime, Zhou [55] considered vertex-transitive covers of non-simple graphs of order 4.To determine the uniqueness of normal quotient graphs of connected pentavalent sym-metric graph of order twice a prime power for some given orders and to compute theirautomorphism groups, covering techniques are employed. In this paper we first provethat these normal quotients are symmetric elementary abelian covers of the dipole Dip and then determine all symmetric elementary abelian covers of Dip , which consist offour infinite families of Cayley graphs on generalized dihedral groups, that is, the graphs CGD p ( p = 5 or 5 | ( p − CGD p (5 | ( p ± CGD p ( p = 5 or 5 | ( p − CGD p . These covers are not isomorphic to each other and their full automorphism groupsare computed. As an application, pentavalent symmetric graphs of order twice a primesquare are classified. 3 Preliminaries
In this section, we describe some preliminary results which will be used later. First wedescribe stabilizers of connected pentavalent symmetric graphs.
Proposition 2.1 [23, Theorem 1.1]
Let X be a connected pentavalent ( G, s ) -transitivegraph for some G ≤ Aut( X ) and s ≥ . Let v ∈ V ( X ) . Then s ≤ and one of thefollowing holds: (1) For s = 1 , G v ∼ = Z , D or D ; (2) For s = 2 , G v ∼ = F , F × Z , A or S , where F is the Frobenius group of order ; (3) For s = 3 , G v ∼ = F × Z , A × A , S × S or (A × A ) ⋊ Z with A ⋊ Z = S and A ⋊ Z = S ; (4) For s = 4 , G v ∼ = ASL(2 , , AGL(2 , , AΣL(2 , or AΓL(2 , ; (5) For s = 5 , G v ∼ = Z ⋊ ΓL(2 , . For a subgroup H of a group G , denote by C G ( H ) the centralizer of H in G and by N G ( H ) the normalizer of H in G . Then C G ( H ) is normal in N G ( H ). Proposition 2.2 [25, Theorem 6.11]
The quotient group N G ( H ) /C G ( H ) is isomorphic toa subgroup of the automorphism group Aut( H ) of H . Let G be a finite group and let π ( G ) = { p | p is a prime divisor of | G |} . Herzog [24]and Shi [52] (also see [26]) classified nonabelian finite simple groups G for | π ( G ) | = 3 and | π ( G ) | = 4 respectively, from which one may deduce the following proposition. Proposition 2.3
Let p ≥ be a prime and let G be a nonabelian simple group. (1) If | π ( G ) | = 3 then G ∼ = A , A , PSL(2 , , PSL(2 , , PSL(2 , , PSL(3 , , PSU(3 , or PSU(4 , with order · · , · · , · · , · · , · · , · · , · · or · · , respectively. (2) Let π ( G ) = { , , , p } with p ≥ . If p | | G | , ∤ | G | and ∤ | G | , then G ∼ =PSL(2 , or PSp(4 , with order · · · or · · · , respectively. Let G be a finite group and S a subset of G with 1 S and S − = S . The Cayleygraph
Γ = Cay(
G, S ) on G with respect to S is defined to have vertex set V (Γ) = G andedge set E (Γ) = {{ g, sg } | g ∈ G, s ∈ S } . It is well-known that Aut(Γ) contains the rightregular representation R ( G ) of G , the acting group of G by right multiplication, and Γ isconnected if and only if G = h S i , that is, S generates G . A Cayley graph Cay( G, S ) is saidto be normal if the right regular representation R ( G ) of G is normal in Aut(Cay( G, S )).By Godsil [21] or Xu [54], we have the following result.4 roposition 2.4
Let
Γ = Cay(
G, S ) be a connected Cayley graph on a finite group G withrespect to S , and let A = Aut(Γ) . Then N A ( R ( G )) = R ( G ) ⋊ Aut(
G, S ) , where N A ( R ( G )) is the normalizer of R ( G ) in A . In particular, Γ is normal if and only if A = Aut( G, S ) . Let p be a prime and let D p = h a, b | a p = b = 1 , b − ab = a − i be the dihedral groupof order 2 p . For p = 5, let ℓ = 1 and for 5 | ( p − ℓ be an element of order 5 in Z ∗ p .Define CD p = Cay( D p , { b, ab, a ℓ +1 b, a ℓ + ℓ +1 b, a ℓ + ℓ + ℓ +1 b } ) . (1)The graph CD p is symmetric because the map α : a a ℓ , b ab induces an automor-phism of D p permuting the elements in { b, ab, a ℓ +1 b, a ℓ + ℓ +1 b, a ℓ + ℓ + ℓ +1 b } cyclicly. By [20,Theorem 3.1], CD p is independent of the choice of ℓ , and by [4] and [20, Theorem 3.1], wehave the following. Proposition 2.5
Let Γ be a connected pentavalent edge-transitive graph of order p fora prime p . Then Γ is symmetric and one of the following holds: (1) X ∼ = K , the complete graph of order and Aut( K ) ∼ = S ; (2) X ∼ = CD ( ∼ = K , ) , the complete bipartite graph of order and Aut( CD ) ∼ = ( S × S ) ⋊ Z ; (3) X ∼ = CD p with | ( p − . For p = 11 , Aut( CD p ) ∼ = PGL(2 , and for p ≥ , Aut( CD p ) ∼ = D p ⋊ Z . Let p be a prime and let D p = h a, b | a p = b = 1 , b − ab = a − i be the dihedral groupof order 2 p . Let 5 | ( p −
1) and ℓ an element of order 5 in Z ∗ p . Define CD p = Cay( D p , { b, ab, a ℓ +1 b, a ℓ + ℓ +1 b, a ℓ + ℓ + ℓ +1 b } ) . (2)Similar to the graph CD p , the graph CD p is symmetric because the map a a ℓ , b ab induces an automorphism of D p . By [20, Theorem 3.1], CD p is independent of the choiceof ℓ , and by [28, Proposition 2.2 and Theorem A], we have the following. Proposition 2.6
Let p be a prime and Γ be a connected pentavalent Cayley graph on D p .If Γ is N Aut (Γ) ( R ( D p )) -arc-transitive, then | ( p − and Γ ∼ = CD p with Aut( CD p ) ∼ = R ( D p ) ⋊ Z . Let Γ be a connected symmetric graph of prime valency and let G ≤ Aut(Γ) be arc-transitive. Let N be a normal subgroup of G . In view of [34, Theorem 9], we have: Proposition 2.7 If N has more than two orbits then the quotient graph Γ N has the samevalency as Γ and N is the kernel of G acting on the set of orbits of N . Furthermore, N is semiregular on V (Γ) and Γ N is G/N -arc-transitive.
For an abelian group H , the generalized dihedral group of H , denoted by GD H , is thesemidirect product H ⋊ Z with the involution of Z inverting every element in H . To endthis section, we consider some special bipartite graphs.5 emma 2.8 Let Γ be a bipartite graph and H an abelian semiregular automorphism groupof Γ with the two bipartite sets of Γ as its orbits. Then Γ is a Cayley graph on GD H . Proof.
Let B and B be the bipartite sets of Γ. Then H acts regularly on each of B and B , and we may assume that B = { h | h ∈ H } and B = { h ′ | h ∈ H } . The actionsof H on B and B are just by right multiplication, that is, h g = hg and ( h ′ ) g = ( hg ) ′ forany h, g ∈ H . Let the neighbors of 1 in Γ be h ′ , h ′ , · · · , h ′ n , where h , h , · · · , h n ∈ H .Since H is abelian, for any h ∈ H , the neighbors of h are ( hh ) ′ , ( hh ) ′ , · · · , ( hh n ) ′ , andfurthermore, the neighbors of h ′ are hh − , hh − , · · · , hh − n . It is easy to check that themap α , defined by h ( h − ) ′ , h ′ h − , h ∈ H , is an automorphism of Γ of order 2.Now for any g, h ∈ H , we have h gα = ( g − h − ) ′ = h αg − and ( h ′ ) gα = g − h − = ( h ′ ) αg − .It follows that gα = αg − , that is, α − gα = g − . Thus, h H, α i = GD H and Γ is a Cayleygraph on GD H . GD Z np with n = 2 , , In this section, we shall construct four families of Cayley graphs on generalized dihedralgroups, and investigate their automorphisms, which will be used later. For a prime p ,write GD p = GD Z p , GD p = GD Z p and GD p = GD Z p for short.Let GD p = h a, d, h | a p = d p = h = [ a, d ] = 1 , h − ah = a − , h − dh = d − i . For p = 5,let ℓ = 1, and for 5 | ( p − ℓ be an element of order 5 in Z ∗ p . Define CGD p = Cay(GD p , { h, ah, a ℓ ( ℓ +1) − d ℓ − h, a ℓ d ( ℓ +1) − h, dh } ) . (3)For 5 | ( p ± λ be an element in Z ∗ p such that λ = 5. Define CGD p = Cay(GD p , { h, ah, a − (1+ λ ) dh, ad − (1+ λ ) h, dh } ) . (4)Let GD p = h a, b, d, h | a p = b p = d p = h = [ a, b ] = [ a, d ] = [ b, d ] = 1 , h − ah = a − , h − bh = b − , h − dh = d − i . For p = 5, let ℓ = 1, and for 5 | ( p − ℓ be anelement of order 5 in Z ∗ p . Define CGD p = Cay(GD p , { h, ah, bh, a − ℓ b − ℓ d − ℓ − h, dh } ) . (5)Let GD p = h a, b, c, d, h | a p = b p = c p = d p = h = [ a, b ] = [ a, c ] = [ a, d ] = [ b, c ] =[ b, d ] = [ c, d ] = 1 , h − ah = a − , h − bh = b − , h − ch = c − , h − dh = d − i . Define CGD p = Cay(GD p , { h, ah, bh, ch, dh } ) . (6) Theorem 3.1
Let
Γ = Cay(
G, S ) be one of the graphs defined in Eqs (3)-(6). Let P be aSylow p -subgroup of R ( G ) and let A = Aut(Γ) . Then Γ is N A ( P ) -arc-transitive. (1) Let
Γ =
CGD p ( p = 5 or | ( p − . If p = 5 then N A ( R ( GD )) ∼ = R ( GD ) ⋊ F and if | ( p − then N A ( R ( GD p )) ∼ = R ( GD p ) ⋊Z . Furthermore, | N A ( P ) | 6 = 20 p . Let
Γ =
CGD p (5 | ( p ± . Then N A ( R ( GD p )) ∼ = R ( GD p ) ⋊ D and | N A ( P ) | hasa divisor p . (3) Let
Γ =
CGD p ( p = 5 or | ( p − . If p = 5 then N A ( R ( GD )) ∼ = R ( GD ) ⋊ S and if | ( p − then R ( GD p ) ⋊ Z ≤ N A ( R ( GD p )) . (4) Let
Γ =
CGD p . Then N A ( R ( GD p )) ∼ = R ( GD p ) ⋊ S . Proof.
By Proposition 2.4, N A ( R ( G )) = R ( G ) ⋊ Aut(
G, S ). For each graph in Eqs (3)-(6), we have h S i = G and | R ( G ) : P | = 2. Thus, Γ is connected and Aut( G, S ) acts faith-fully on S , implying Aut( G, S ) ≤ S . Furthermore, P ⊳ R ( G ) and hence P ⊳ N A ( R ( G )),forcing that N A ( R ( G )) ≤ N A ( P ). If (1)-(4) are true, then Γ is N A ( P )-arc-transitivebecause it is N A ( R ( G ))-arc-transitive. To finish the proof, it suffices to show (1)-(4).Let Γ = CGD p = Cay( G, S ). Then p = 5 or 5 | ( p − G =GD p = h a, d, h | a p = d p = h = [ a, d ] = 1 , h − ah = a − , h − dh = d − i and S = { h, ah, a ℓ ( ℓ +1) − d ℓ − h, a ℓ d ( ℓ +1) − h, dh } , where ℓ = 1 for p = 5 and ℓ is an element of order 5in Z ∗ p for 5 | ( p − ℓ = 1, ℓ + ℓ + ℓ + ℓ + 1 = 0 and ( ℓ + 1) − = − ℓ − ℓ in the field Z p . It is easy to check that α : h ah, a a ℓ ( ℓ +1) − − d ℓ − , d a − inducesan automorphism of order 5 of GD p permuting the elements in S cyclicly.For p = 5, the map β : a ad , d a d and h h , induces an automorphism of order4 of GD p permuting the elements in { ah, ad h, dh, a dh } cyclicly. Take γ ∈ Aut(
G, S )such that h γ = h and ( ah ) γ = ah . Then { a d, ad , d } γ = { a d, ad , d } . If d γ = a d or ad then ( a d ) d = ad or a d , which is impossible because ad, a d
6∈ { a d, ad , d } . Thus, d γ = d and hence γ = 1, implying | Aut(
G, S ) | = 20. By Lemma 2.1, N A ( R ( GD )) ∼ = R ( GD ) ⋊ F . Since N A ( R ( G )) ≤ N A ( P ), we have | N A ( P ) | 6 = 20 p .For 5 | ( p − δ ∈ Aut(
G, S ) such that h δ = h . Then δ fixes { a, a ℓ ( ℓ +1) − d ℓ − , a ℓ d ( ℓ +1) − , d } setwise, and hence fixes a · a ℓ ( ℓ +1) − d ℓ − · a ℓ d ( ℓ +1) − · d , that is, a ℓ +2 ℓ +2 d ℓ + ℓ +2 .Thus, ( a r d rℓ ) δ = a r d rℓ , where r = ℓ +2 ℓ +2. Suppose r = 0. Then 2 ℓ +2 ℓ +1 = ℓ r = 0and 3 ℓ + 4 ℓ − ℓ = 2(2 ℓ + 2 ℓ + 1) − r = 0, that is, 3 ℓ + 4 ℓ − ℓ + ℓ + 2 ℓ = 2( ℓ + ℓ + ℓ + ℓ + 1) − r = 0, that is, 2 ℓ + ℓ + 2 = 0. It followsthat 5 ℓ ( ℓ + 1) = (3 ℓ + 4 ℓ −
2) + (2 ℓ + ℓ + 2) = 0, and hence p = 5, a contradiction. Thisimplies that ( ad ℓ ) δ = ad ℓ . One may compute the following equations. a (1 − ℓ (1+ ℓ ) − ) = ( ad ℓ ) · ( a ℓ ( ℓ +1) − d ℓ − ) − , a − = ( ad ℓ ) · ( a ℓ d ( ℓ +1) − ) − ℓ ( ℓ +1) ,a − ℓ ( ℓ +1) − = ( ad ℓ )( a ℓ ( ℓ +1) − d ℓ − ) − , a (1 − ℓ ) = ( ad ℓ )( ad ) − ℓ ,a ( ℓ + ℓ ) = ( ad ℓ )( a ℓ d ( ℓ +1) − ) − ℓ ( ℓ +1) , a = ( ad ℓ ) d − ℓ . ( ∗ )Recall that δ fixes S = { a, a ℓ ( ℓ +1) − d ℓ − , a ℓ d ( ℓ +1) − , d } setwise. If δ fixes a then it alsofixes d because d = ( ad ℓ ) ℓ a − ℓ , which implies that δ = 1. By Eq( ∗ ), if δ fixes any elementin S then δ fixes a and therefore δ = 1.Suppose that a δ = a . Then either h δ i has two orbits of length 2, or it is transitive on S . For the former, δ fixes one element in { a · a ℓ ( ℓ +1) − d ℓ − , a · a ℓ d ( ℓ +1) − , a · d } , that is, { a ℓ ( ℓ +1) − d ℓ − , a ℓ d ( ℓ +1) − , ad } . By Eq( ∗ ), δ fixes a , a contradiction. For the latter, δ has two orbits of length 2 on S , and hence δ fixes one element in { a · a ℓ ( ℓ +1) − d ℓ − , a · ℓ d ( ℓ +1) − , a · d } = { a ℓ ( ℓ +1) − d ℓ − , a ℓ d ( ℓ +1) − , ad } . Again by Eq( ∗ ), δ fixes a and δ = 1,which is impossible because δ is transitive on S .Since δ = 1, we have that | Aut(
G, S ) | = 5, N A ( R ( GD p )) ∼ = R ( GD p ) ⋊ Z and | N A ( R ( GD p )) | = 10 p . Suppose | N A ( P ) | = 20 p . Since N A ( R ( GD p )) ≤ N A ( P ), we have | N A ( P ) : N A ( R ( GD p )) | = 2, implying N A ( R ( GD p )) ⊳ N A ( P ). Note that R ( GD p ) is ahall { , p } -subgroup of N A ( R ( GD p )). Then R ( GD p ) is characteristic in N A ( R ( GD p ))and hence R ( GD p ) ⊳ N A ( P ). It follows that N A ( P ) ≤ N A ( R ( GD p )), which is impossible.Thus, | N A ( P ) | 6 = 20 p , completing the proof of (1).Let Γ = CGD p = Cay( G, S ). Then 5 | ( p ± G = GD p = h a, d, h | a p = d p = h = [ a, d ] = 1 , h − ah = a − , h − dh = d − i and S = { h, ah, a − (1+ λ ) dh, ad − (1+ λ ) h, dh } ,where λ = 5 in the field Z p . The map α : h ah, a a − (1+ λ ) − d, d a − induces anautomorphism of GD p permuting the elements in S cyclicly. Take β ∈ Aut(
G, S ) suchthat h β = h . Then β fixes { a, a − (1+ λ ) d, ad − (1+ λ ) , d } setwise and hence a · a − (1+ λ ) d · ad − (1+ λ ) · d , that is, ( ad ) − (1+ λ ) . If 2 + 2 − (1 + λ ) = 0 then λ = −
5, which is impossiblebecause λ = 5. Thus, β fixes ad .Note that β fixes { a, a − (1+ λ ) d, ad − (1+ λ ) , d } setwise. Clearly, if β fixes a then β = 1.Assume that a β = a . Then a β = a − (1+ λ ) d, ad − (1+ λ ) or d . If a β = a − (1+ λ ) d then ad = ( ad ) β = a − (1+ λ ) dd β , that is, d β = a − − (1+ λ ) . This is impossible because d β a, a − (1+ λ ) d, ad − (1+ λ ) , d } . Thus, a β = a − (1+ λ ) d , and similarly, a β = ad − (1+ λ ) becauseotherwise d β = d − − (1+ λ ) . It follows that a β = d and d β = ( a − ad ) β = d − ad = a , thatis, β is an automorphism of order 2 of GD p induced by a d , d a and h h . Thisimplies that the subgroup of Aut( G, S ) fixing h is h β i . Thus, | Aut(
G, S ) | = 10 and byProposition 2.1, N A ( R ( GD p )) ∼ = R ( GD p ) ⋊ D . Since N A ( R ( G )) ≤ N A ( P ), we have that | N A ( P ) | has a divisor 20 p . This completes the proof of (2).Let Γ = CGD p = Cay( G, S ). Then p = 5 or 5 | ( p − G = GD p = h a, b, d, h | a p = b p = d p = h = [ a, b ] = [ a, d ] = [ b, d ] = 1 , h − ah = a − , h − bh = b − , h − dh = d − i and S = { h, ah, bh, a − ℓ b − ℓ d − ℓ − h, dh } , where ℓ = 1 for p = 5 and ℓ is an element of order 5 in Z ∗ p for 5 | ( p − α : h ah, a ba − , b a − ℓ − b − ℓ d − ℓ − , d a − induces an automorphism of GD p permuting the elements in S cyclicly. Thus, R ( GD p ) ⋊ Z ≤ N A ( R ( GD p )). For p = 5, the map β : a b , b d , d a − b − d − and h h , induces an automorphism of order 4 of GD per-muting the elements in { ah, bh, dh, a − b − d − h } cyclicly. Furthermore, any permutationon { a, b, d } with h h induces an automorphism of GD . Thus, Aut( G, S ) ∼ = S and N A ( R ( GD )) ∼ = R ( GD ) ⋊ S . This completes the proof of (3).Let Γ = CGD p = Cay( G, S ). Then S = { h, ah, bh, ch, dh } and G = GD p = h a, b, c, d, h | a p = b p = c p = d p = h = [ a, b ] = [ a, c ] = [ a, d ] = [ b, c ] = [ b, d ] = [ c, d ] =1 , h − ah = a − , h − bh = b − , h − ch = c − , h − dh = d − i . The map α : h ah, a ba − , b ca − , c da − , d a − induces an automorphism of GD p permuting the ele-ments in S cyclicly. Furthermore, any permutation on { a, b, c, d } with h h induces anautomorphism of GD p . Thus, Aut( G, S ) ∼ = S and N A ( R ( GD p )) ∼ = R ( GD p ) ⋊ S . Thiscompletes the proof of (4). 8 Symmetric elementary abelian covers of
Dip and compute their automorphism groups by combining Theorem 3.1.An epimorphism P : e Γ Γ of graphs is called a regular covering projection or regular N -covering projection if Aut( e Γ) has a semiregular subgroup N whose orbits on V ( e Γ)coincide with the vertex fibres P − ( v ) , v ∈ V (Γ), and the arc and edge orbits of N coincidewith the arc fibres P − (( u, v )) , u ∼ v , and edge fibres P − ( { u, v } ) , u ∼ v , respectively.In particular, we call the graph e Γ a regular cover or an N -cover of the graph Γ, and N the covering transformation group . In particular, if N is a cyclic or an elementary abeliangroup, then we speak of e Γ as a cyclic cover or an elementary abelian cover of Γ. Let P : e Γ Γ be a regular covering projection. An automorphism of e Γ is said to be fibre-preserving if it maps a vertex fibre to a vertex fibre, and all such fibre-preserving automorphismsform a group, say F , called the fibre preserving group . When e Γ is connected, it is easyto show that N is the kernel of Aut( e Γ) acting on the fibres and F = N Aut ( e Γ) ( N ). If e Γ is F -arc-transitive, we say that e Γ is a symmetric cover of Γ.Two regular covering projections P : e Γ Γ and P ′ : e Γ ′ Γ of a graph Γ areisomorphic if there exist an automorphism α ∈ Aut(Γ) and an isomorphism e α : e Γ e Γ ′ such that α P = P ′ e α . If α is identity, then P and P ′ are equivalent , and if e Γ = e Γ ′ and P = P ′ then we call e α a lift of α and α a projection of e α along P .Let Γ be a graph and let N be a finite group. Assign to each arc ( u, v ) of Γ a voltage ζ ( u, v ) ∈ N such that ζ ( u, v ) = ζ ( u, v ) − , where ζ : Γ N is called a voltage assignment of Γ. Let Cov (Γ; ζ ) be the derived graph from ζ , which has vertex set V (Γ) × N andadjacency relation defined by ( u, a ) ∼ ( v, aζ ( u, v )), where a ∈ N and u ∼ v in Γ. Theprojection onto the first coordinate P : Cov (Γ; ζ ) Γ is a regular N -covering projection,where the group N acts semiregularly via left multiplication on the second coordinate of( u, a ), u ∈ V (Γ), a ∈ N . Give a spanning tree T of Γ, a voltage assignment ζ is said to be T-reduced if the voltages on the tree arcs are identity. Gross and Tucker [22] showed thatevery regular covering of a graph Γ can be derived from a T -reduced voltage assignment ζ with respect to an arbitrary fixed spanning tree of Γ. It is clear that if ζ is reduced,then the derived graph Γ × ζ N is connected if and only if the voltages on the cotree arcsgenerate the voltage group N .A voltage assignment on arcs can be extended to a voltage assignment on walks ina natural way. Given α ∈ Aut(Γ), we define a function α from the set of voltages onfundamental closed walks based at a fixed vertex v ∈ V (Γ) to the voltage group N by( ζ ( C )) α = ζ ( C α ), where C ranges over all fundamental closed walks at v , and ζ ( C ) and ζ ( C α ) are the voltages of C and C α , respectively. Clearly, if N is abelian, then α doesnot depend on the choice of the base vertex, and the fundamental closed walks at v canbe substituted by the fundamental cycles generated by the cotree arcs of Γ. The nextproposition is a special case of Theorem 4.2 in [36].9 roposition 4.1 Let P : e Γ =
Cov (Γ; ζ ) Γ be a regular N -covering projection. Thenan automorphism α of Γ lifts if and only if α extends to an automorphism of N . By [38, Corollary 3.3(a)], we have the following proposition.
Proposition 4.2
Let P : Cov (Γ; ζ ) Γ and P : Cov (Γ; ζ ) Γ be two regular N -covering projections of a graph Γ . Then P and P are isomorphic if and only if there isan automorphism δ ∈ Aut(Γ) and an automorphism η ∈ Aut( N ) such that ( ζ ( W )) η = ζ ( W δ ) for all fundamental closed walks W at some base vertex of Γ . By Theorem 3.1 (5),
CGD p ( p = 5 or 5 | ( p − CGD p (5 | ( p ± CGD p ( p = 5or 5 | ( p − CGD p are symmetric elementary abelian covers of Dip . Note that allthese graphs have girth 6. Theorem 4.3
Let p be a prime and Z np an elementary abelian group with n ≥ . Let Γ be a connected symmetric Z np -cover of the dipole Dip . Then ≤ n ≤ and (1) For n = 2 , Γ ∼ = CGD p ( p = 5 or | ( p − or CGD p (5 | ( p ± , which are uniquefor a given order; Aut(
CGD ) = ( R ( GD ) ⋊ F ) Z ∼ = Z · (( F × F ) ⋊ Z ) with N A ( R ( GD )) = R ( GD ) ⋊ F , Aut(
CGD p ) = R ( GD p ) ⋊ Z for | ( p − , and Aut(
CGD p ) = R ( GD p ) ⋊ D ; (2) For n = 3 , Γ ∼ = CGD p ( p = 5 or | ( p − , which are unique for a given order; Aut(
CGD ) = R ( GD ) ⋊ S and Aut(
CGD p ) = R ( GD p ) ⋊ Z for | ( p − ; (3) For n = 4 , Γ ∼ = CGD p and Aut(
CGD p ) = R ( GD p ) ⋊ S . Proof.
Let A = Aut(Γ). Let N = Z np and G = N A ( N ). Then N has two orbits and issemiregular on V (Γ). By hypothesis, Γ is G -arc-transitive. Clearly, Γ N is Dip and itsvertices are denoted by u and v that are connected by five multiple edges (see Fig. 1). a b c d v u a a a a a Figure 1: Dip with a voltage assignment ζ .Let P : Γ = Dip × ζ Z np Dip be the corresponding covering projection, where ζ isits voltage assignment. Since N ⊳ G , the projection L of G is an arc-transitive subgroupof Aut(Dip ). Thus, L lifts to G and G/N ∼ = L . Furthermore, L is the largest subgroup ofAut(Dip ) which can be lifted along P . Label the five arcs of Dip starting from u by a , a , a , a and a , respectively. Let T be a spanning tree of Dip corresponding to the arc10 . We may assume that P is T -reduced. Write ζ ( a ) = 1, ζ ( a ) = a , ζ ( a ) = b , ζ ( a ) = c and ζ ( a ) = d . Since Γ is connected, N = Z np = h a, b, c, d i , forcing that n ≤ α = ( a a a a a )( a − a − a − a − a − ), β = ( a a a a )( a − a − a − a − ), γ = ( a a − )( a a − )( a a − )( a a − )( a a − ), δ = ( a a a )( a − a − a − ), and ε =( a a a a )( a − a − a − a − ). Then α, β, γ, δ, ε ∈ Aut(Dip ). There are four fundamentalcycles: W = a a − , W = a a − , W = a a − and W = a a − . We list all these cyclesand their voltages in Table 1, in which ζ ( W i ) denotes the voltage on W i . W i ζ ( W i ) W αi ζ ( W αi ) W βi ζ ( W βi ) W β i ζ ( W β i ) W = a a − a a a − ba − a a − ca − a a − bc − W = a a − b a a − ca − a a − a − a a − ac − W = a a − c a a − da − a a − ba − a a − c − W = a a − d a a − a − a a − da − a a − dc − W i ζ ( W i ) W γi ζ ( W γi ) W δi ζ ( W δi ) W εi ζ ( W εi ) W = a a − a a − a a − a a − ba − a a − da − W = a a − b a − a b − a a − a − a a − ba − W = a a − c a − a c − a a − ca − a a − a − W = a a − d a − a d − a a − da − a a − ca − Table 1: Voltages on fundamental cycles and images under α , β , β , γ , δ , ε Clearly, Aut(Dip ) = S ⋊ Z , where S fixes u and v . By the arc-transitivity of L , | L | is divisible by 10. Let L ∗ be the subgroup of L fixing u and v . Then | L : L ∗ | = 2. Thus, L ∗ ≤ S and a Sylow 5-subgroup of L ∗ is also a Sylow 5-subgroup of Aut(Dip ). SinceSylow 5-subgroups of Aut(Dip ) are conjugate, we may assume α ∈ L , that is, α lifts.Noting that α β = α , we have that L ∗ = h α i , h α, β i , h α, β i , A or S . In particular, if β cannot lift then L ∗ = h α i ∼ = Z ; if β lifts but β and δ cannot then L ∗ = h α, β i ∼ = D ; if β lifts but δ cannot then L ∗ ∼ = Z ⋊ Z ; if α , β and δ lift then L ∗ = S .Consider the mapping ¯ α from the set of voltages on the four fundamental cycles of Dip to the elementary abelian group Z np , defined by ζ ( W i ) ¯ α = ζ ( W αi ), 1 ≤ i ≤
4. Similarly,we may define ¯ β , ¯ β , ¯ γ , ¯ δ and ¯ ε . By Table 1, a ¯ γ = a − , b ¯ γ = b − , c ¯ γ = c − , and d ¯ γ = d − .Since Z np is abelian, ¯ γ can be extended to an automorphism of Z np . By Proposition 4.1, γ lifts along P . This implies that L lifts if and only if L ∗ lifts.Since α lifts, by Proposition 4.1, ¯ α can be extended to an automorphism of Z np , denotedby α ∗ . Again by Table 1, a α ∗ = ba − , b α ∗ = ca − , c α ∗ = da − , d α ∗ = a − . (7)Suppose b = 1. By Eq (7), ca − = b α ∗ = 1, that is, c = a . Thus, d = b = 1 because c α ∗ = a α ∗ , and hence c − = a − = d α ∗ = 1. It follows that Z np = h a, b, c, d i = 1, acontradiction. Thus, b = 1. Similarly, a = 1, c = 1 and d = 1. Since n ≤
4, we have N = Z p , Z p or Z p .Suppose d ∈ h a i . Then d = a k , k ∈ Z ∗ p . By Eq (7), a − = b k a − k , implying b ∈ h a i .Similarly, c ∈ h a i . It follows that Z p = h a, b, c, d i = h a i , a contradiction. Thus, d / ∈ h a i .11 ase 1: . N = Z p .Since d / ∈ h a i , we have Z p = h a, d i . Let b = a i d j and c = a ℓ d m for some i, j, ℓ, m ∈ Z p .By Eq (7), a i ( i − − j d ij = ( a i − d j ) i a − j = ( ba − ) i a − j = ( a i d j ) α ∗ = b α ∗ = ca − = a ℓ − d m and a ( i − ℓ − m d jℓ = ( a i − d j ) ℓ a − m = ( ba − ) ℓ a − m = ( a ℓ d m ) α ∗ = c α ∗ = a − d . Then the followingequations hold in the field Z p . i ( i − − j = ℓ − ,ij = m, ( i − ℓ − m = − ,jℓ = 1 . (8)(9)(10)(11)By Eqs (11) and (9), ℓ = 0, j = ℓ − and i = mℓ . By Eq (10), ℓ ( mℓ − − m = − ℓ − mℓ + m −
1) = 0. Thus, ℓ − mℓ + m − Z p .Assume ℓ = 1. Then j = ℓ − = 1 and i = mℓ = m . By Eq (8), i ( i −
1) = 1, whichimplies that p = 2 and (2 i − = 5. By elementary number theory, p = 5 k ± p = 5.Let p = 5 k ±
1. Since (2 i − = 5, we have i = 2 − (1 + λ ), where λ = 5. It followsthat b = a − (1+ λ ) d and c = ad − (1+ λ ) . Note that 2 − (1 + λ ) − − − (1 − λ ). ByTable 1, ¯ α can be extended to the automorphism of Z p induced by a a − − (1 − λ ) d and d a − , and ¯ β can be extended to the automorphism induced by a a − − (1 − λ ) d − (1 − λ ) and d a − d − (1 − λ ) . Furthermore, ¯ β and ¯ δ cannot be extended to automorphisms of Z p .Thus, L ∗ ∼ = D and | G | = | N || L | = 20 p .We claim that Γ is unique for any prime p such that p = 5 k ±
1. This is sufficientto show that Γ is independent of the choice of λ . Note that the equation x = 5 hasexactly two roots in Z p , that is, ± λ . Since (2 i − = 5, we have i = 2 − (1 ± λ ).It follows that b = a − (1 ± λ ) d and c = ad − (1 ± λ ) . Clearly, the voltage assignment ζ is determined by ζ ( a ), ζ ( a ), ζ ( a ), ζ ( a ) and ζ ( a ), and for convenience, write ζ =( ζ ( a ) , ζ ( a ) , ζ ( a ) , ζ ( a ) , ζ ( a )). It follows that ζ = ζ = (1 , a, a − (1+ λ ) d, ad − (1+ λ ) , d ) or ζ = ζ = (1 , a, a − (1 − λ ) d, ad − (1 − λ ) , d ). By Table 1, ζ ( W ) = a , ζ ( W ) = a − (1+ λ ) d , ζ ( W ) = ad − (1+ λ ) , ζ ( W ) = d and ζ ( W β ) = d − (1 − λ ) , ζ ( W β ) = a − , ζ ( W β ) = a − (1 − λ ) − d , ζ ( W β ) = da − . Let η be the automorphism of Z p induced by a d − (1 − λ ) and d da − . Then ( ζ ( W i )) η = ζ ( W βi ) for each 1 ≤ i ≤
4, and by Proposition 4.2,Dip × ζ Z p ∼ = Dip × ζ Z p , as claimed.Let p = 5. Then (2 i − = 0 implies i = 3 in Z . Thus, m = i = 3. It follows that b = a d and c = ad . By Table 1, ¯ α and ¯ β can be extended to the automorphisms of Z p induced by a a d , d a − , and a d , d da − , respectively. But, ¯ δ can not beextended to an automorphism of Z p . Thus, L ∗ ∼ = Z ⋊ Z and | G | = 40 × .Assume mℓ + m − ℓ + 1 = 0 and m = ( ℓ + 1) − . Recall that j = ℓ − and i = mℓ . Then i = ℓ ( ℓ + 1) − . By Eq (8), ℓ ( ℓ + 1) − ( ℓ ( ℓ + 1) − − − ℓ − = ℓ − ℓ + ℓ + ℓ + ℓ + 1 = 0. Thus, ℓ − ℓ − ℓ + ℓ + ℓ + ℓ + 1) = 0. For p = 5, we have ℓ = 1 because ℓ = 1 in Z ∗ , which has been discussed in the previousparagraph. Let p >
5. Then ℓ is an element of order 5 in Z ∗ p and hence 5 | ( p − ℓ + 1) − = − ℓ − ℓ and ( ℓ + 1) − = − ℓ − ℓ . Note that b = a ℓ ( ℓ +1) − d ℓ − and12 = a ℓ d ( ℓ +1) − . By Table 1, ¯ α can be extended to the automorphism of Z p induced by a a ℓ ( ℓ +1) − − d ℓ − and d a − , but ¯ β cannot. Thus, L ∗ ∼ = Z and | G | = 10 p .We now claim that Γ is unique for any prime p with 5 | ( p − Z ∗ p hasexactly four elements of order 5, that is, ℓ i for 1 ≤ i ≤
4. By the arbitrariness of ℓ , itsuffices to show that Dip × ζ Z p ∼ = Dip × ζ Z p , where ζ and ζ are voltage assignmentscorresponding to ℓ and ℓ respectively, that is, ζ = (1 , a, a ℓ ( ℓ +1) − d ℓ − , a ℓ d ( ℓ +1) − , d ) and ζ = (1 , a, a ℓ ( ℓ +1) − d ℓ − , a ℓ d ( ℓ +1) − , d ). Recall that ( ℓ + 1) − = − ℓ − ℓ and ( ℓ + 1) − = − ℓ − ℓ . By Table 1, ζ ( W ) = a , ζ ( W ) = a ℓ + ℓ +1 d ℓ , ζ ( W ) = a ℓ d ℓ + ℓ +1 , ζ ( W ) = d ,and ζ ( W ε ) = a − d , ζ ( W ε ) = a ℓ + ℓ d ℓ , ζ ( W ε ) = a − , ζ ( W ε ) = a ℓ − d ℓ + ℓ +1 . Let η be the automorphism of Z p induced by a a − d and d a ℓ − d ℓ + ℓ +1 . It is easy tocheck that ( ζ ( W i )) η = ζ ( W εi ) for each 1 ≤ i ≤
4, and by Proposition 4.2, Dip × ζ Z p ∼ =Dip × ζ Z p , as claimed.We have proved that for p = 5, the graph Γ is unique with | N A ( N ) | = 40 × ; for5 | ( p − | N A ( N ) | = 10 p ; for 5 | ( p ± | N A ( N ) | = 20 p . By Theorem 3.1 (1) and (2), the uniqueness of Γ impliesthat Γ ∼ = CGD p ( p = 5 or 5 | ( p − CGD p (5 | ( p ± ∼ = CGD p ( p = 5 or 5 | ( p − | Aut(
CGD ) | = 2 · andAut( CGD ) has a normal subgroup of order 5. By Theorem 3.1 and Proposition 2.1, N A ( R ( GD )) = R ( GD ) ⋊ F and A = ( R ( GD ) ⋊ F ) Z ∼ = Z · (( F × F ) ⋊ Z ),where ( F × F ) ⋊ Z is a maximal solvable subgroup of Aut( K , ) ∼ = ( S × S ) ⋊ Z . For5 | ( p − | Aut(
CGD p ) | = 10 p . Again by MAGMA [6], | Aut(
CGD p ) | =10 p for p = 11 , , , ,
71. Assume p ≥
91. Since | N A ( N ) | = 10 p , it suffices to show N ⊳ A . Suppose N ⋪ A . Since A = R ( GD p ) A w for any w ∈ V (Γ), N is a Sylow p -subgroup of A . By Sylow Theorem, the number of Sylow p -subgroups of A is kp + 1 and kp + 1 = | A : N A ( N ) | . It follows that kp + 1 = | A w | / N A ( N ) = R ( GD p ) ⋊ Z byTheorem 3.1. Since CGD p has girth 6, Γ is at most 3-arc-transitive and by Proposition 2.1, | A w | | · ·
5. Thus, kp + 1 is a divisor of 2 · . Since p ≥
91 and 5 | ( p − k, p ) = (1 , kp = 2 · −
1. However, this is impossible by Proposition 2.1because | A w | = 5( kp + 1) = 2 · ·
5. Thus, | Aut(
CGD p ) | = 10 p and A = R ( GD p ) ⋊ Z .Let Γ ∼ = CGD p (5 | ( p ± | Aut(
CGD p ) | = 20 p . By MAGMA [6], thisis true for p = 11 , , , , , , ,
71. Assume p ≥
79 and it suffices to show that N ⊳ A because N A ( N ) = R ( GD p ) ⋊ D . Suppose N ⋪ A . As the previous paragraph,the number kp + 1 of Sylow p -subgroups of A is a divisor of | A w | /
10 = 2 · . However,no such a prime p exists for p ≥
79. Thus, | Aut(
CGD p ) | = 20 p and A = R ( GD p ) ⋊ D . Case 2: N = Z p .Recall that d
6∈ h a i . Suppose b ∈ h a, d i . Then b = a i d j , i, j ∈ Z p . By Eq (7), ca − = b α ∗ = ( b i a − i ) a − j , implying c ∈ h a, d i . It follows that Z p = h a, b, c, d i = h a, d i , acontraction. Thus, b / ∈ h a, d i and Z p = h a, d, b i . Let c = a i b j d k for some i, j, k ∈ Z p . Againby Eq (7), da − = c α ∗ = ( a i b j d k ) α ∗ = ( ba − ) i ( a i − b j d k ) j ( a − ) k = a ij − i − j − k b i + j d kj , and thefollowing equations hold in the field Z p . 13 j − i − j − k = − ,i + j = 0 ,kj = 1 . (12)(13)(14)By Eqs (13) and (14), i = − j and k = j − . By Eq (12), − j + j − j − j − + 1 = 0,that is, j − j + j − j + 1 = 0. Let ℓ = − j . Then ℓ + ℓ + ℓ + ℓ + 1 = 0 and ℓ − ℓ − ℓ + ℓ + ℓ + ℓ + 1) = 0. It follows that either p = 5 with ℓ = 1, or5 | ( p −
1) with ℓ an element of order 5 in Z ∗ p .Let p = 5. Then c = a b d . By Table 1, ¯ α can be extended to the automorphism of Z p induced by a a b , b a b d , and d a . Thus, Γ is symmetric.Let 5 | ( p −
1) with ℓ an element of order 5 in Z ∗ p . Then c = a − ℓ b − ℓ d − ℓ . By Table 1,¯ α can be extended to the automorphism of Z p induced by a a − b , b a − ℓ − b − ℓ d − ℓ ,and d a − . Thus, Γ is symmetric. Further, ¯ β cannot be extended to an automorphismof Z p . Thus, L ∗ ∼ = Z and | G | = 10 p . We now claim that Γ is unique. Since Z ∗ p hasexactly four elements of order 5, that is, ℓ i for each 1 ≤ i ≤
4, it suffices to show thatDip × ζ Z p ∼ = Dip × ζ Z p , where ζ and ζ are voltage assignments corresponding to ℓ and ℓ respectively, that is, ζ = (1 , a, b, a − ℓ b − ℓ d − ℓ , d ) and ζ = (1 , a, b, a − ℓ b − ℓ d − ℓ , d ). ByTable 1, ζ ( W ) = a , ζ ( W ) = b , ζ ( W ) = a − ℓ b − ℓ d − ℓ , ζ ( W ) = d , and ζ ( W ε ) = a − d , ζ ( W ε ) = a − b , ζ ( W ε ) = a − , ζ ( W ε ) = a − − ℓ b − ℓ d − ℓ . Let η be the automorphism of Z p induced by a a − d , b a − b, d a − − ℓ b − ℓ d − ℓ − . Then ( ζ ( W i )) η = ζ ( W εi ) foreach 1 ≤ i ≤
4, and by Proposition 4.2, Dip × ζ Z p ∼ = Dip × ζ Z p , as claimed.We have proved that p = 5 or 5 | ( p − | N A ( N ) | = 10 p . Then Theorem 3.1 (3) implies that Γ ∼ = CGD p . By MAGMA [6], | Aut(
CGD ) | = 2 · · and | Aut(
CGD ) | = 2 · · , and again by Theorem 3.1 (3),Aut( CGD ) = R ( GD ) ⋊ S and Aut( CGD ) = R ( GD ) ⋊ Z . Assume p ≥
31. Weclaim that | Aut(
CGD p ) | = 10 p . It suffices to show N ⊳ A because | N A ( N ) | = 10 p .Suppose to the contrary that N ⋪ A . Since A = R ( GD p ) A w for any w ∈ V (Γ), N is aSylow p -subgroup of A . By Sylow Theorem, the number of Sylow p -subgroups in A is kp +1and kp + 1 = | A : N A ( N ) | . Since N A ( N ) = R ( GD p ) ⋊ Z , we have kp + 1 = | A w | /
5, andsince
CGD p has girth 6, it is at most 3-arc-transitive. By Proposition 2.1, | A w | | · · kp + 1 is a divisor of 2 · . Since p ≥
31 and 5 | ( p − k, p ) = (1 , kp = 2 −
1, ( k, p ) = (1 , kp = 2 · −
1, ( k, p ) = (1 ,
71) and kp = 2 · − k, p ) = (7 ,
41) and kp = 2 · −
1. It follows that | A w | = 2 ·
5, 2 · ·
5, 2 · · · ·
5. Since Γ is not 4-arc-transitive, by Proposition 2.1 only the last case canhappen and in this case, p = 41 and Γ ∼ = CGD is 3-arc-transitive. However, it is easyto check by MAGMA [6] that there is no 6-cycle in CGD passing through the 3-arc(1 , h, ah · h, h · ah · h ) = (1 , h, a, a − h ) (one may take ℓ = 10 because 10 = 1 in Z ),which is impossible because of the 3-arc-transitivity of Γ. Thus, | Aut(
CGD p ) | = 10 p for5 | ( p −
1) and Aut(
CGD p ) = R ( GD p ) ⋊ Z . Case 3: N = Z p .Let N = h a i × h b i × h c i × h d i . By the connectedness of Γ, we may let ζ ( a ) = 1,14 ( a ) = a , ζ ( a ) = b , ζ ( a ) = c , and ζ ( a ) = d . By Table 1, it is easy to see that ¯ α , ¯ β and ¯ δ can be extended to automorphisms of Z p . Thus, L ∗ = S and | G | = 240 p . ByTheorem 3.1, Γ ∼ = CGD p .For p = 2 , , | Aut(
CGD p ) | = 2 · · · p . This isalso true for any prime p ≥
11. To prove it, we only need to show that N ⊳ A because | G | = | N A ( N ) | = 2 · · · p . Suppose to the contrary that N ⋪ A . Note that N isa Sylow p -subgroup because A = R ( GD p ) A w for any w ∈ V (Γ). By Sylow Theorem,the number of Sylow p -subgroups in A is kp + 1 and kp + 1 = | A : N A ( N ) | . Since N A ( N ) = R ( GD p ) ⋊ S , we have kp + 1 = | A w | / CGD p has girth 6, it is atmost 3-arc-transitive. By Proposition 2.1, | A w | | · · kp + 1 is a divisor of2 ·
3. Since p ≥
11, we have ( k, p ) = (1 ,
11) and kp = 2 · −
1, or ( k, p ) = (1 ,
23) and kp = 2 · −
1. It follows that | A w | = 2 · · · ·
5, and by Proposition 2.1, Γ is 3-arc-transitive. Note that Γ ∼ = CGD or CGD . It is easy to check that there is no 6-cyclein CGD and CGD passing through the 3-arc (1 , h, ah · h, h · ah · h ) = (1 , h, a, a − h ),which contradicts the 3-arc-transitivity of Γ. Thus, | Aut(
CGD p ) | = 240 p for each prime p , and by Theorem 3.1 (4), Aut( CGD p ) = R ( GD p ) ⋊ S . In this section we investigate pentavalent symmetric graphs of order 2 p n and basic onesof such graphs are determined. We first prove the following lemma. Lemma 5.1
Let p be a prime and let Γ be a connected pentavalent symmetric graph oforder p n with n ≥ . Let G be an arc-transitive subgroup of Aut(Γ) . Then every minimalnormal subgroup of G is an elementary abelian p -group. Proof.
Let u ∈ V (Γ). By Proposition 2.1, | G u | | · ·
5, and hence | G | | · · · p n .Let N be a minimal normal subgroup of G . Then N ∼ = T m for a finite simple group T .Suppose that N is nonabelian. If N has more than two orbits, by Proposition 2.7, | N | | p n and hence N is solvable, a contradiction. Thus, N has one or two orbits on V (Γ), which implies that p n | | N | . Since | G u | | · · | V (Γ) | = 2 p n , we have | G | = 2 i · j · · p n with 1 ≤ i ≤
10 and 0 ≤ j ≤
2. Thus, π ( N ) = 4 or 3, where π ( N ) isthe number of distinct prime factors of | N | .If π ( N ) = 4 then p ≥ ∤ | G | , N = T is a simple { , , , p } -group suchthat 2 ∤ | N | , 3 ∤ | N | . Suppose that p | | N | . By Proposition 2.3, we have 5 | | N | and hence 5 | | G | , a contradiction. Thus, p ∤ | N | , and since p n | | N | , we have n = 1,contrary to the hypothesis. It follows that π ( N ) = 3. Since Γ has the prime valency 5, G u is primitive on Γ ( u ), the neighborhood of u in Γ, and since N u ⊳ G u , either N u = 1or 5 | | N u | . If N u = 1 then N is semiregular on V (Γ). Thus, | N | | p n and hence N issolvable, a contradiction. Thus, 5 | | N u | , and by Proposition 2.3, N is a { , , } -group.Furthermore, p = 5 , T ∼ = A , A or PSU(4 , p = 5, we have N = T or T as 3 ∤ | G | . It follows that 5 ∤ | N | , and since N hasat most two orbits and 5 | | N u | , we have n = 1, contrary to the hypothesis.15or p = 2 or 3, we have N = T as 5 ∤ | G | . If p = 2 then N = A or A as 3 ∤ | G | ,which implies that | Γ | = 8 or 16 because 2 n | | N | . By McKay [35], there is no pentavalentsymmetric graph of order 8 and there is a unique pentavalent symmetric graph of order16 that is the graph F Q , but this is impossible because Aut( F Q ) ∼ = Z ⋊ S has nonormal subgroup A or A . Thus, p = 3. Since | Γ | = 2 · n ≥
18 and since there isno pentavalent symmetric graph of order 18 by McKay [35], we have N =PSU(4 , | PSU(4 , | = 2 · ·
5, we have | Γ | = 2 · or 2 · . If N is transitive on V (Γ)then | N u | = 2 · · ·
5, which is impossible by Proposition 2.1. Therefore, N hastwo orbits and | N u | = 2 · · ·
5. Clearly, Γ is bipartite and G has a 2-element, say g , interchanging the two bipartition sets of Γ. It follows that H = N h g i is arc-transitiveand | H u | = 2 i +6 · ·
5, or 2 i +6 · i ≥
0. Again by Proposition 2.1, H u ∼ = ASL(2 , , N u ⊳ H u , we have N u ∼ = ASL(2 , ,
2) has nosubgroup isomorphic to ASL(2 ,
4) by MAGMA [6], a contradiction.Now N is abelian and hence elementary abelian. Since | V (Γ) | = 2 p n , N is a 2-groupor a p -group. If p = 2 and N is a 2-group then the quotient graph Γ N has odd order andvalency 5, a contradiction. Thus, N is a p -group. This completes the proof.The hypercube Q n is the Cayley graph Cay( Z n , { a , a , · · · , a n } ), where Z n = h a i ×h a i × · · · × h a n i , and it is well-known that Aut( Q n ) = R ( Z n ) ⋊ S n . The folded hyper-cube F Q n ( n ≥
2) is the Cayley graph
F Q n = Cay( Z n , S ), where S = { a , a , · · · , a n ,a a · · · a n } . It is easy to see that Aut( Z n , S ) is transitive on S and any permutation on { a , a , · · · , a n } induces an automorphism of Z n , which fixes S setwise. This implies that S n +1 ≤ Aut( Z n , S ). For n ≥ F Q n has a unique 4-cycle passing through 1 and any twoelements in S , and hence Aut( F Q n ) = R ( Z n ) ⋊ S n +1 . Theorem 5.2
Let p be a prime and let Γ be a connected pentavalent symmetric graph oforder p n with n ≥ . Then Γ is a normal cover of one of the following graphs: K , F Q , CD p ( p = 5 or | ( p − , CGD p ( p = 5 or | ( p − , CGD p (5 | ( p ± , CGD p ( p = 5 or | ( p − , or CGD p . Proof.
Let A = Aut(Γ) and let M be a maximal normal subgroup of A which has morethan two orbits on V (Γ). By Proposition 2.7, Γ is a normal cover of the quotient graphΓ M , which has valency 5 and is A/M -arc-transitive. Clearly, | V (Γ M ) | = 2 p m for a positiveinteger m . We aim to show that Γ M ∼ = K , F Q , CD p ( p = 5 or 5 | ( p − CGD p ( p = 5or 5 | ( p − CGD p (5 | ( p ± CGD p ( p = 5 or 5 | ( p − CGD p .Let N be a minimal normal subgroup of A/M . If m = 1 then, by Proposition 2.5,Γ M ∼ = K , or CD p ( p = 5 or 5 | ( p − m ≥
2. ByLemma 5.1, N is an elementary abelian p -group, and by the maximality of M , N has oneor two orbits on V (Γ M ).First assume that N has one orbit, that is, N is transitive on V (Γ M ). Then N actsregularly on V (Γ M ) and | N | = | V (Γ M ) | . It follows that N is an elementary abelian 2-group and Γ M = Cay( N, S ). Since Γ M has valency 5, the connectedness of Γ M impliesthat | N | ≤
32. Clearly, if | N | = 32 then Γ M ∼ = Q = CGD . By McKay [35], there isno pentavalent symmetric graph of order 8 and there is a unique pentavalent symmetricgraph of order 16. Thus, if | N | = 16 then Γ M ∼ = F Q .16ow assume that N has two orbits on V (Γ M ). Then Γ M is a bipartite graph with thetwo orbits of N as its bipartite sets. Let u ∈ V (Γ M ). If N u = 1 then Γ M ∼ = K , , contraryto the assumption m ≥
2. Thus, N is regular on each bipartite set of Γ M and hence Γ M is a symmetric N -cover of the dipole Dip . By Theorem 4.3, Γ M ∼ = CGD p ( p = 5 or5 | ( p − CGD p (5 | ( p ± CGD p ( p = 5 or 5 | ( p − CGD p . Theorem 5.3
Let p be a prime and n a positive integer. The basic graphs of connectedpentavalent symmetric graph of order p n are K , F Q , CGD , CD p ( p = 5 or | ( p − , CGD p (5 | ( p ± or CGD p ( p = 3 or p ≥ . Proof.
Let Γ = K , F Q , CGD , CD p ( p = 5 or 5 | ( p − CGD p (5 | ( p ± CGD p ( p = 3 or p ≥ A = Aut(Γ) and N a nontrivial normal subgroup of A .The graphs K , F Q and CGD are basic because N has one or two orbits byMAGMA [6]. Since there is no pentavalent graph of odd order, by Proposition 2.7, N is transitive on V (Γ) or each orbit of N on V (Γ) has odd length. In particular, CD p ( p = 5 or 5 | ( p − CGD p (5 | ( p ± | N | 6 = p . By MAGMA [6],Aut( CGD ) has no normal subgroups of order 11, and we may assume that p > | N | = p . By Theorem 4.3, A = Aut( CGD p ) = R ( GD p ) ⋊ D , and hence A/N has stabilizer D on V (Γ N ), which is impossible for p >
11 by Proposition 2.5. It followsthat
CGD p (5 | ( p ± CGD p ( p = 3 or p ≥ p , p or p (for p = 11 by MAGMA [6] andfor p = 3 , p ≥
13 by Theorem 4.3 and Proposition 2.5), and hence
CGD p ( p = 3 or p ≥
7) are basic.On the other hand, let Γ =
CGD p ( p = 5 or 5 | ( p − CGD p (5 | ( p − CGD p ( p = 2 or 5) and let A = Aut(Γ). To finish the proof, by Theorem 5.2 and Proposition 2.7,it suffices to show that A has a nontrivial normal subgroup having more than two orbits.This is true for Γ = CGD , CGD or CGD by MAGMA [6], and moreover, Γ is anormal cover of CD , F Q or CGD , respectively. Let Γ = CGD p (5 | ( p − A = Aut( CGD p ) = R ( GD p ) ⋊ Z and A u ∼ = Z with u ∈ V (Γ). The group R ( GD p )contains a characteristic subgroup of order p , say Z p , and each subgroup of Z p is normalin R ( GD p ). The number of subgroups of order p in Z p is p + 1, and since 5 ∤ ( p + 1), atleast one of the subgroups of order p in Z p is fixed by A u . It follows that A has a normalsubgroup of order p and hence Γ is a normal cover of CD p (5 | ( p − CGD p (5 | ( p − R ( GD p ) has a characteristic subgroup Z p , which has p + p + 1subgroups of order p . Since 5 ∤ ( p + p + 1), A has a normal subgroup of order p and Γis a normal cover of CD p (5 | ( p − p p . 17 heorem 6.1 Let p be a prime and Γ a connected pentavalent symmetric graph of order p . Then Γ ∼ = CGD p ( p = 5 or | ( p − , CGD p (5 | ( p ± or CD p (5 | ( p − . Proof.
Let A = Aut(Γ) and u ∈ V (Γ). By McKay [35], there is no pentavalent symmetricgraph of order 8 or 18. Thus, p ≥
5. By Proposition 2.1, | A u | | · · | A | = 2 i · j · · p , 1 ≤ i ≤
10, 0 ≤ j ≤ Observation: If p = 5 then A has a non-abelian Sylow 5-group.Suppose that Q is an abelian Sylow 5-subgroup of A . Since | V (Γ) | = 2 · , we have | Q | = 5 and | Q u | = 5. Thus, Q has two orbits, say B and B with u ∈ B . Since Q is abelian, Q u fixes every vertex in B and all orbits of Q u in B have length 5. Thus,Γ contains a subgroup K , , and by its connectedness, Γ ∼ = K , , which is contrary to | Γ | = 2 p . The observation follows.Let H be a non-trivial normal abelian p -group of an arc-transitive subgroup B of A .Then all orbits of H have length p or p because | Γ | = 2 p . Suppose H u = 1. Since H u ⊳ B u , we have 5 | | H u | and p = 5. By Proposition 2.7, H has two orbits on V (Γ),implying that 5 | | H | . Consequently, 5 | | H | and H is an abelian Sylow 5-subgroup of A ,which is impossible by Observation. Thus, H is semiregular on V (Γ), forcing that | H | = p or p . This implies that H = Z p , Z p or Z p . If H = Z p then, by Theorem 4.3, Γ ∼ = CGD p ( p = 5 or 5 | ( p − CGD p (5 | ( p ± Assumption:
Each non-trivial normal abelian p -group of each arc-transitive subgroup of A is semiregular on V (Γ) and isomorphic to Z p or Z p .Let N be a minimal normal subgroup of A . By Lemma 5.1 and Assumption, N ∼ = Z p .Now we prove the following claim. Claim: A has a semiregular subgroup L of order p such that N ≤ L and Γ is N A ( L )-arc-transitive.By Proposition 2.7, the quotient graph Γ N has order 2 p and is A/N -arc-transitive. ByProposition 2.5, Γ N ∼ = CD ( ∼ = K , ) with Aut( CD ) ∼ = ( S × S ) ⋊ Z , Γ N ∼ = CD withAut( CD ) ∼ = PGL(2 , N ∼ = CD p ( p >
11 and 5 | ( p − CD p ) ∼ = D p ⋊ Z .Since A/N is arc-transitive, in each case
A/N contains a semiregular subgroup of order p ,say L/N . Thus, 10 p | | A/N | , N ≤ L and L is a semiregular subgroup of order p in A .For p >
11 with 5 | ( p − CD p ) ∼ = D p ⋊ Z . Since 10 p | | A/N | , wehave A/N = Aut( CD p ) and hence L is the unique normal Sylow p -subgroup of A . Thus, N A ( L ) = A and Γ is N A ( L )-arc-transitive. For p = 11, Aut( CD ) = PGL(2 ,
11) and
L/N is a Sylow 11-subgroup of Aut( CD ). By ATLAS [5], Aut( CD ) has a maximal subgroup M/N ∼ = Z ⋊ Z such that L/N ≤ M/N . Thus, L ⊳ M . Since M/N PSL(2 , ≤ PGL(2 , M/N is vertex-transitive and hence arc-transitive on Γ N . It follows that M is arc-transitive on Γ and Γ is N A ( L )-arc-transitive. For p = 5, Γ is a bipartite graphof order 5 with the orbits of L as its bipartite sets. By Lemma 2.8, N A ( L ) is vertex-transitive. Since Sylow 5-subgroups of A have order 5 and | L | = p , N A ( L ) contains aSylow 5-subgroup of A , and hence Γ is N A ( L )-arc-transitive.18ow we are ready to finish the proof. Set B = N A ( L ). By Claim, Γ is B -arc-transitive.Since L ⊳ B and L has two orbits, Γ is bipartite with its bipartite sets as the orbits of L ,say B and B with u ∈ B . By Assumption, L ∼ = Z p . Note that L is a Sylow p -subgroupof C B ( L ) for p > p ∤ | A | , where C B ( L ) is the centralizer of L in B . This is alsotrue for p = 5 by Observation. Thus, L is a normal Sylow p -subgroup of C B ( L ) and hence C B ( L ) = L × K , where K is a Hall p ′ -subgroup of C B ( L ). In particular, K is characteristicin C B ( L ). If K = 1, then K E B because C B ( L ) E B , and since p ∤ | K | , K has more thantwo orbits on V (Γ). By Proposition 2.7, K is semiregular and hence | K | = 2. But thequotient graph Γ K would have odd order p and valency 5, a contradiction. Thus, K = 1and C B ( L ) = L . Since B/L = B/C B ( L ) . Aut( L ) ∼ = Z p ( p − , B has a unique normalsubgroup of order 2 p containing L , say R . By Lemma 2.8, B contains a regular dihedralgroup of order 2 p containing L . By the uniqueness of R , we have R ∼ = D p and hence Γis a Cayley graph on R . Since R ⊳ B , Proposition 2.6 implies Γ ∼ = CD p (5 | ( p − Acknowledgement:
This work was supported by the National Natural Science Founda-tion of China (11571035, 11231008, 11271012) and by the 111 Project of China (B16002).
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