https://doi.org/10.26493/1855-3974.1894.37c (Also available at http://amc-journal.eu)
Arc-transitive graphs of valency twice a prime admit a semiregular automorphism ∗
Michael Giudici
†Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia
Gabriel Verret
‡Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
Received 1 January 2019, accepted 16 February 2020, published online 15 October 2020
Abstract
We prove that every finite arc-transitive graph of valency twice a prime admits a non- trivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant Conjecture, which states that all finite vertex-transitive digraphs admit such automorphisms.
Keywords: Arc-transitive graphs, polycirculant conjecture, semiregular automorphism.
Math. Subj. Class. (2020): 20B25, 05E18
1 Introduction
All graphs in this paper are finite. In 1981, Maruˇsiˇc asked if every vertex-transitive digraph admits a nontrivial semiregular automorphism [13], that is, an automorphism whose cycles all have the same length. This question has attracted considerable interest and a generali- sation of the affirmative answer is now referred to as the Polycirculant Conjecture [4]. See [1] for a recent survey on this problem.
One line of investigation of this question has been according to the valency of the graph or digraph. Every vertex-transitive graph of valency at most four admits such an automor- phism [7,14], and so does every vertex-transitive digraph of out-valency at most three [9].
∗Authors are grateful to the anonymous referees for their helpful suggestions.
†The research of the first author was supported by the ARC Discovery Project DP160102323.
‡Gabriel Verret is grateful to the N.Z. Marsden Fund for its support (via grant UOA1824).
E-mail addresses:[email protected] (Michael Giudici), [email protected] (Gabriel Verret)
cbThis work is licensed under https://creativecommons.org/licenses/by/4.0/
Every arc-transitive graph of prime valency has a nontrivial semiregular automorphism [10]
and so does every arc-transitive graph of valency8[17]. Partial results were also obtained for arc-transitive graphs of valency a product of two primes [18]. We continue this theme by proving the following theorem.
Theorem 1.1. Arc-transitive graphs of valency twice a prime admit a nontrivial semiregu- lar automorphism.
2 Preliminaries
IfGis a group of automorphisms of a graphΓandvis a vertex ofΓ, we denote byGvthe stabiliser inGofv, byΓ(v)the neighbourhood ofv, and byGΓ(v)v the permutation group induced byGvonΓ(v). We will need the following well-known results.
Lemma 2.1. LetΓbe a connected graph andG6Aut(Γ). If a primepdivides|Gv|for somev∈V(Γ), then there existsu∈V(Γ)such thatpdivides|GΓ(u)u |.
Proof. Sincepdivides|Gv|, there exists an elementgof orderpinGv. Asg 6= 1, there are vertices not fixed byg. Among these vertices, letwbe one at minimal distance fromv.
LetP be a path of minimal length fromvtowand letube the vertex precedingwonP.
By the definition ofw, we have thatuis fixed byg, sog ∈Gu. On the other hand,gdoes not fix the neighbourwofu, sogΓ(u)6= 1hence|gΓ(u)|=pand the result follows.
Lemma 2.2. LetGbe a permutation group and letK be a normal subgroup ofGsuch thatG/Kacts faithfully on the set ofK-orbits. IfG/Khas a semiregular elementKgof orderrcoprime to|K|, thenGhas a semiregular element of orderr.
Proof. See for example [17, Lemma 2.3].
Lemma 2.3. A transitive group of degree a power of a prime pcontains a semiregular element of orderp.
Proof. In a transitive group of degree a power of a primep, every Sylowp-subgroup is transitive. A non-trivial element of the center of this subgroup must be semiregular.
Recall that a permutation group isquasiprimitiveif every non-trivial normal subgroup is transitive, andbiquasiprimitiveif it is transitive but not quasiprimitive and every non- trivial normal subgroup has at most two orbits.
3 Arc-transitive graphs of prime valency
In the most difficult part of our proof, the arc-transitive graph of valency twice a prime will have a normal quotient with prime valency. We will thus need a lot of information about arc-transitive graphs of prime valency, which we collect in this section. We start with the following result, which is [3, Theorem 5]:
Theorem 3.1. LetΓbe a connectedG-arc-transitive graph of prime valencypsuch that the action ofGonV(Γ)is either quasiprimitive or biquasiprimitive. Then one of the following holds:
(1) Gcontains a semiregular element of odd prime order;
(2) |V(Γ)|is a power of 2;
(3) Γ = K12,G= M11andp= 11;
(4) |V(Γ)|= (p2−1)/2sandG= PSL2(p)orPGL2(p), wherepis a Mersenne prime andsis a proper divisor of(p−1)/2but also a multiple of the product of the distinct prime divisors of(p−1)/2;
(5) |V(Γ)|= (p2−1)/sandG= PGL2(p), wherepandsare as in part (4), andΓis the canonical double cover of the graph given in (4).
(Recall that thecanonical double coverof a graphΓisΓ×K2.) We note that in cases (4) and (5), we must havep>127, since this is the smallest Mersenne primepsuch that (p−1)/2is not squarefree. This fact will be used at the end of Section4.
Corollary 3.2. LetΓbe a connectedG-arc-transitive graph of prime valency. Then one of the following holds:
(1) Gcontains a semiregular element of odd prime order;
(2) |V(Γ)|is a power of 2;
(3) Gcontains a normal2-subgroupPsuch that(Γ/P, G/P)is one of the graph-group pairs in (3–5) of Theorem3.1.
Proof. Suppose that|V(Γ)|is not a power of2. IfGis quasiprimitive or biquasiprimitive onV(Γ), then the result follows immediately from Theorem 3.1 (withP = 1in case (3)). We thus assume that this is not the case and letP be a normal subgroup of Gthat is maximal subject to having at least three orbits onV(Γ). In particular,P is the kernel of the action ofGon the set ofP-orbits. HenceG/P acts faithfully, and quasiprimitively or biquasiprimitively onV(Γ/P). SinceΓ has prime valency, is connected and G-arc- transitive, [12, Theorem 9] implies thatP is semiregular. We may thus assume thatP is a2-group. (OtherwiseP contains a semiregular element of odd prime order.) IfG/P contains a semiregular element of odd prime order, then Lemma2.2implies that so does G. We may assume that this is not the case. Similarly, we may assume that|V(Γ/P)|is not a power of2. (Otherwise,|V(Γ)|is a power of2.) It follows thatΓ/P andG/P are as is (3–5) of Theorem3.1.
We will now prove some more results about the graphs that appear in (3–5) of Theo- rem3.1. Let us first recall the notion ofcoset graphs. LetGbe a group with a subgroup H and letg ∈ Gsuch thatg2 ∈ H butg /∈ NG(H). The graphCos(G, H, HgH)has vertices the right cosets ofH inG, with two cosets HxandHyadjacent if and only if xy−1 ∈HgH. Observe that the action ofGon the set of vertices by right multiplication induces an arc-transitive group of automorphisms such thatH is the stabiliser of a vertex.
Moreover, every arc-transitive graph can be constructed in this way [16, Theorem 2].
Lemma 3.3. The graphs in (3) and (4) of Theorem3.1have a3-cycle.
Proof. ClearlyK12has a3-cycle so suppose thatΓis one of the graphs given in (4). Let Gbe as in Theorem3.1and letv ∈ V(Γ). ThenGis one ofPSL2(p)orPGL2(p)and acts arc-transitively onΓ. In both cases, letX = PSL2(p), soG =X or|G : X| = 2.
By [3, Lemma 5.3], we have thatGv ∼= Cp oCs ifG = PSL2(p), and Cp oC2sif
G= PGL2(p). By [5, pp. 285–286],PGL2(p)has a unique conjugacy class of subgroups of orderpand the normaliser of such a subgroup is isomorphic toCpoCp−1, which is the stabiliser inPGL2(p)of a1-dimensional subspace of the natural2-dimensional vector space. The intersection of this subgroup withX is isomorphic toCpoC(p−1)/2, which has odd order. It follows that if|G:X|= 2, thenGvis not contained inX. Thus, in both cases,X is transitive onV(Γ). SinceX is normal inG,Xv 6= 1andΓhas prime valency, it follows thatXis arc-transitive onΓand soΓ = Cos(X, H, HgH)whereH ∼= CpoCs andg∈X\NX(H)such thatHgHis a union ofpdistinct right cosets ofH.
SinceHhas a characteristic subgroupY of orderp, it follows thatNX(H)normalises Y and so NX(H) 6 NX(Y) ∼= CpoC(p−1)/2. SinceNX(Y)/Y is cyclic it follows thatNX(H) = NX(Y). Also note thatNX(H)is the stabiliser inX of a1-dimensional subspace and so the action ofX on the set of right cosets ofNX(H)is2-transitive and the stabiliser of any two1-dimensional subspaces is isomorphic toC(p−1)/2. Letx∈X.
The stabiliser inX of the cosetHx isHx and so the orbit ofHx underH has length
|H :H ∩Hx|. In particular,H fixes the cosetHxif and only ifx∈NX(H), and soH fixes|NX(H) : H| = (p(p−1)/2)/ps = (p−1)/2spoints ofV(Γ). Moreover, since the stabiliser of two1-dimensional subspaces is isomorphic toC(p−1)/2 it follows that if x /∈NX(H)thenH∩Hx∼= C(p−1)/2and so|H :H∩Hx|=p. Thus the points ofV(Γ) that are not fixed byHare permuted byH in orbits of sizepand so for anyg /∈NX(H) we have thatHgHis a union ofpdistinct right cosets ofH.
For eachx∈NX(H)define the bijectionλxofV(Γ)byHy 7→ x−1Hy =Hx−1y.
SinceXacts onV(Γ)by right multiplication, we see thatλxcommutes with each element ofX. Moreover,λxis nontrivial if and only ifx /∈H. LetC ={λx | x∈NG(H)} 6 Sym(V(Γ)). Since X acts transitively onV(Γ)andCcentralisesX, it follows from [6, Theorem 4.2A] thatCacts semiregularly onV(Γ). Now|C|=|NX(H) :H|= (p−1)/2s andX×C 6Sym(V(Γ)). SinceC EX×C, the set of orbits ofCforms a system of imprimitivity forX×Cand hence forX. Moreover, sinceC is semiregular, comparing orders yields that C has |V(Γ)|/|C| = p+ 1 orbits. One of these orbits is the set of fixed points ofHandHtransitively permutes the remainingporbits ofC. In particular, it follows thatCtransitively permutes the nontrivial orbits ofHand so the isomorphism type ofΓdoes not depend on the choice of the double cosetHgH.
LetZ be the subgroup of scalar matrices inSL2(p)and letHˆ be the subgroup of the stabiliser inSL2(p)of the1-dimensional subspaceh(1,0)isuch thatH = ˆH/Z. Note that ˆh =
1 0
1 1
∈ Hˆ and letgˆ =
0 1
−1 0
. In particular, lettingh = ˆhZ andg = ˆgZ we have thatg /∈ NX(H)and so we may assume thatΓ = Cos(X, H, HgH). NowHg andHghare both adjacent toH and one can check thatg(gh)−1 =hgh∈HgHand so {H, Hg, Hgh}is a3-cycle inΓ.
Definition 3.4. LetΓbe a graph and letS0be a subset ofV(Γ). LetS=S0. If a vertexu outsideShas at least two neighbours inS, addutoS. Repeat this procedure until no more vertices outsideS have this property. If at the end of the procedure, we haveS = V(Γ), then we say thatΓisdense with respect toS0.
It is an easy exercise to check that denseness is well-defined.
Corollary 3.5. LetΓbe a graph in (3) or (4) of Theorem3.1and letS0 ={u, v}be an edge of Γ. ThenΓis dense with respect toS0.
Proof. SinceΓis arc-transitive of prime valencyp, the local graph atv(that is, the sub- graph induced onΓ(v)) is a vertex-transitive graph of order pand thus vertex-primitive.
By Lemma3.3, Γhas a3-cycle so the local graph has at least one edge and thus must be connected. It follows that, running the process described in Definition3.4starting at S=S0, eventuallySwill contain all neighbours ofv. Repeating this argument and using connectedness ofΓyields the desired conclusion.
The following is immediate from Definition3.4.
Lemma 3.6. LetΓbe a graph andS0be a set of vertices such thatΓis dense with respect toS0. Then the canonical double cover of Γ, with vertex-setV(Γ)× {0,1}, is dense with respect toS0× {0,1}.
Proof. LetSi be the sequence of subsets of V(Γ)obtained when running the procedure from Definition3.4starting withS0and ending withSnfor somen. SinceΓis dense with respect toS0, we haveSn= V(Γ). Fori∈ {1, . . . , n}, letvi=Si\Si−1. (In other words, v1is the first vertex added toS0to getS1, thenv2is added toS1to getS2, etc.)
Now, letΓ0 = Γ×K2be the canonical double cover ofΓand letS00=S0× {0,1} ⊆ V(Γ0). We now run the procedure from Definition3.4starting atS00. At the first step, we note that, sincev1was added toS0, it must have at least two neighbours inS0, sayu1and w1. It follows that both(v1,0) and(v1,1) also have at least two neighbours inS00 (for example,(u1,1)and(w1,1), and(u1,0)and(w1,0), respectively). We thus add(v1,0) and(v1,1)toS00to getS10 =S00 ∪ {(v1,0),(v1,1)}. Note thatS10 =S1× {0,1}. We then repeat this procedure, preserving the conditionSi0 =Si× {0,1}at each iteration. At the end of this process, we haveSn0 = Sn× {0,1} = V(Γ)× {0,1} = V(Γ0)and soΓ0 is dense with respect toS0× {0,1}.
4 Proof of Theorem 1.1
Letpbe a prime, letΓ be an arc-transitive graph of valency 2pand let G = Aut(Γ).
We may assume thatΓis connected. IfGis quasiprimitive or bi-quasiprimitive, thenG contains a nontrivial semiregular element, by [8, Theorem 1.2] and [10, Theorem 1.4]. We may thus assume thatGhas a minimal normal subgroupN such thatN has at least three orbits. In particular,Γ/Nhas valency at least2.
IfNis nonabelian, thenGhas a nontrivial semiregular element by [18, Theorem 1.1].
We may therefore assume thatN is abelian and, in particular,Nis an elementary abelian q-group for some primeq.
We may also assume thatN is not semiregular that is,Nv 6= 1for some vertexv. It follows from Lemma2.1that1 6=NvΓ(v)EGΓ(v)v . AsΓisG-arc-transitive, we have that GΓ(v)v is transitive and so the orbits ofNvΓ(v)all have the same size, either2orp. SinceN is aq-group, this size is equal toq. Writingdfor the valency ofΓ/N, we have that either (d, q) = (2, p)or(d, q) = (p,2).
Ifd = 2andq =p, then it follows from [15, Theorem1] thatΓis isomorphic to the graph denoted byC(p, r, s)in [15]. By [15, Theorem2.13],Aut(C(p, r, s))contains the nontrivial semiregular automorphism%defined in [15, Lemma2.5].
We may thus assume thatd=pandq= 2. In this case, ifuis adjacent tov, thenuhas exactly2 = 2p/dneighbours invN. LetKbe the kernel of the action ofGonN-orbits.
By the previous observation, the orbits ofKvΓ(v) have size2and thus it is a2-group. It follows from Lemma2.1thatKvis a2-group and thus so isK=N Kv.
Now, G/K is an arc-transitive group of automorphisms ofΓ/N, so we may apply Corollary3.2. IfG/Khas a semiregular element of odd prime order, then so doesG, by Lemma2.2. If|V(Γ/N)|is a power of2, then so is|V(Γ)|and, in this case,Gcontains a semiregular involution by Lemma2.3. We may thus assume that we are in case (3) of Corollary3.2, that is,G/K contains a normal2-subgroupP/Ksuch that(Γ/P, G/P)is one of the graph-group pairs in (3–5) of Theorem3.1. Note thatPis a2-group. LetMbe a minimal normal subgroup ofGcontained in the centre ofP. Note thatMis an elementary abelian 2-group. We may assume thatM is not semiregular henceMv 6= 1 and so by Lemma2.1,MvΓ(v)6= 1. Moreover,|M| 6= 2as otherwiseMv=M and we would deduce thatM fixes each element ofV(Γ), a contradiction. SinceM is central in P,Mv fixes every vertex invP.
Note that the G-conjugates ofMv must cover M, otherwise M contains a nontriv- ial semiregular element. By the previous paragraph, the number of conjugates ofMv is bounded above by the number ofP-orbits, that is|V(Γ/P)|, so we have
|M|6|Mv||V(Γ/P)|.
Since Γ is connected and G-arc-transitive, there are no edges within P-orbits. As MvΓ(v) 6= 1, there existsg ∈ Mv such thatwandwg are distinct neighbours ofv. Let ube the other neighbour ofwinvP. SinceMvfixes every element ofvP it follows that uis also a neighbour ofwandwgand so{v, w, u, wg}is a4-cycle inΓ. Thus the graph induced between adjacentP-orbits is a union ofC4’s.
Ifxis a vertex andyP is aP-orbit adjacent tox, then there is a uniqueC4containing xbetweenxP andyP, and thus a unique vertexzantipodal toxin thisC4. We say thatz is thebuddy ofxwith respect toyP. The set of buddies ofvis equal toΓ2(v)∩vP, which is clearly fixed setwise byGv. Moreover, each vertex has the same number of buddies.
Furthermore, sinceGvtransitively permutes the set ofp P-orbits adjacent tovP, eitherv has a unique buddy or it has exactlypbuddies.
Ifv has a unique buddyz, thenΓ(v) = Γ(z), and so swapping every vertex with its unique buddy is a nontrivial semiregular automorphism. Thus it remains to consider the case wherevhaspbuddies. We first prove the following.
Claim. IfX is a subgroup ofM that fixes pointwise bothaP andbP, andcP is aP-orbit adjacent to bothaP andbP, thenXfixescP pointwise.
Proof. Suppose that somex∈Xdoes not fixc. NowxfixesaPpointwise, socxmust be the buddy ofcwith respect toaP. Similarly,cxmust be the buddy ofcwith respect tobP. These are distinct, which is a contradiction. It follows thatX fixescand, sinceX 6M, alsocP.
Lets > 1, letα = (v0, . . . , vs)be ans-arc ofΓ and letα0 = (v0, . . . , vs−1). Now
|vsMvs−1
|= 2, so|Mvs−1 :Mvs−1vs|= 2and|Mα0 :Mα|62. Applying induction yields that
|Mv0 :Mα|62s. (4.1)
We first assume thatΓ/PandG/Pare as in (3) or (4) of Theorem3.1. Let{u, v}be an edge ofΓ. By the previous paragraph, we have|Mv :Muv|62. Recall thatMvfixes all vertices invP, soMuvfixes all vertices invP∪uP. Combining the claim with Corollary3.5
yields thatMuv = 1and thus|Mv|= 2. It follows that|M|6|Mv||V(Γ/P)|so|M|6 2|V(Γ/P)|. SinceM is minimal normal inG, it is an irreducibleG-module overGF(2), of dimension at least two. In fact, sinceM is central inP, it is also an irreducible(G/P)- module. SinceG/P is nonabelian simple or has a nonabelian simple group as an index two subgroup, this implies thatM is a faithful irreducible(G/P)-module overGF(2). If G/P = M11, then|M| > 210[11, Theorem 8.1], contradicting|M| 6 2·12 = 24. If G/P = PSL2(p)or PGL2(p) then by [2, Section VIII], |M| > 2(p−1)/2. Recall that p>127and so this contradicts|M|62(p2−1)/2s < p2−1.
Finally, we assume thatΓ/P is in (5) of Theorem3.1, that is,Γ/P is the canonical double cover of a graphΓ0which appears in (4) of Theorem3.1. In particular,V(Γ/P) = V(Γ0)× {0,1}. By Lemma3.3,Γ0 has a3-cycle, say(u, v, w). By Corollary 3.5and Lemma3.6,Γ/P is dense with respect to{u, v} × {0,1}. Now, let
α= ((u,0),(v,1),(w,0),(u,1),(v,0)).
Sinceαcontains{u, v} × {0,1},Γ/Pis dense with respect toα. Note thatαis a4-arc of Γ/P. Letαbe a4-arc ofΓthat projects toα. SinceΓ/Pis dense with respect toα, arguing as in the last paragraph yieldsMα = 1. On the other hand, ifv ∈V(Γ)is the the initial vertex ofα, then by (4.1), we have|Mv : Mα| 624and thus|Mv| 624. Since|M| 6
|Mv||V(Γ/P)|it follows that|M|624(p2−1)/s. As above,M is a faithful irreducible (G/P)-module overGF(2)of dimension at least two. SinceG/P = PGL2(p)we have from [2] that|M|>2(p−1)/2, which again contradicts|M|624(p2−1)/s <24(p2−1).
ORCID iDs
Michael Giudici https://orcid.org/0000-0001-5412-4656 Gabriel Verret https://orcid.org/0000-0003-1766-4834
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