An explicit characterization of arc-transitive circulants

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Abstract

A reductive characterization of arc-transitive circulants was given independently by Kovács in 2004 and the first author in 2005. In this paper, we give an explicit characterization of arc-transitive circulants and their automorphism groups. Based on this, we give a proof of the fact that arc-transitive circulants are all CI-digraphs.

Introduction

Throughout this paper, a digraph is an ordered pair (V,A) with vertex set V and arc set A, where A is a set of ordered pairs of elements of V. The cardinality of the vertex set is called the order of the digraph. A digraph Γ is said to be arc-transitive if its automorphism group Aut(Γ) acts transitively on the arc set. A digraph Γ is called a circulant if Aut(Γ) has a finite cyclic subgroup that is regular on the vertex set, and is called a normal circulant if Aut(Γ) has a finite cyclic subgroup that is normal and regular on the vertex set.

Since the work of Chao and Wells [5], [6] in the 1970's, considerable effort has been made to characterize arc-transitive circulants in the literature. Remarkable results were achieved under certain conditions such as 2-arc-transitivity [1], [20], square-free order [18], odd prime-power order [23], small valency [2] and unit connection set [2]. Based on Schur ring and permutation group techniques, a reductive characterization for connected arc-transitive non-normal circulants was obtained independently by Kovács [13] and the first author [17]. Recently, a classification of arc-transitive circulants that are 2-distance-transitive was given in [4].

In this paper, we give an explicit characterization (Theorem 1.1) of connected arc-transitive circulants, which reveals their structures and determines their automorphism groups. In fact, Theorem 1.1 shows that a finite connected arc-transitive circulant can be decomposed into a normal circulant, some complete graphs and an edgeless graph by tensor and lexicographic products.

For digraphs Γ and Σ, their tensor product (direct product) is denoted by Γ×Σ, and their lexicographic product is denoted by Γ[Σ] (see Section 2 for the definitions of these products). For a positive integer n, denote by Sn the symmetric group of degree n, and denote by Kn and Kn the complete graph and the edgeless graph of order n, respectively. Let C4 denote an undirected cycle of length 4.

Theorem 1.1

For every connected arc-transitive circulant Γ of order n, there exist a connected arc-transitive normal circulant Γ0 of order n0 and positive integers n1,,nr,b, where r0, such that the following hold:

  • (1)

    Γ0C4;

  • (2)

    ni4 for i=1,,r;

  • (3)

    n=n0n1nrb, and n0,n1,,nr are pairwise coprime;

  • (4)

    Γ(Γ0×Kn1××Knr)[Kb];

  • (5)

    Aut(Γ)Sb(Aut(Γ0)×Sn1××Snr).

Moreover, Γ is uniquely determined by the triple (Γ0,{n1,,nr},b) satisfying the above conditions.

In view of Theorem 1.1 we give the following definition.

Definition 1.2

For a connected arc-transitive circulant Γ, a triple (Γ0,{n1,,nr},b) of a finite connected arc-transitive normal circulant Γ0, a (not necessarily nonempty) set {n1,,nr} of integers and a positive integer b satisfying conditions (1)–(5) in Theorem 1.1 is called a tensor-lexicographic decomposition of Γ.

Here are some remarks on Theorem 1.1:

  • (I)

    The single loop is connected and arc-transitive. On the other hand, if a connected arc-transitive digraph has order at least two, then it has no loop due to the arc-transitivity. Thus the circulant Γ0 in the statement of Theorem 1.1 is a single loop if n0=1 and has no loop if n02. In the former case, the tensor product of Γ0 with any digraph is isomorphic to the digraph, and hence the digraph Γ described in part (3) of Theorem 1.1 is isomorphic to (Kn1××Knr)[Kb].

  • (II)

    For each arc-transitive circulant Γ0 of order n0 and positive integers n1,,nr,b such that n0,n1,,nr are pairwise coprime, the digraph (Γ0×Kn1××Knr)[Kb] is an arc-transitive circulant by Lemma 2.1, Lemma 2.2. The condition ni4 as in (2) of Theorem 1.1 ensures that Kni is a non-normal circulant. If Γ0C4, then since C4K2[K2], we deduce from Lemma 2.3, Lemma 2.4 that (C4×Kn1××Knr)[Kb](K2×Kn1××Knr)[K2b].

  • (III)

    To be precise, the statement “Γ is uniquely determined by the triple (Γ0,{n1,,nr},b)” means that if (Γ0,{n1,,nr},b) and (Σ0,{m1,,ms},c) are two tensor-lexicographic decompositions of Γ then Γ0Σ0, {n1,,nr}={m1,,ms} and b=c. Theorem 1.1 shows that every connected arc-transitive circulant has a unique tensor-lexicographic decomposition.

  • (IV)

    Further descriptions of Γ0 in Theorem 1.1 can be found in Section 3 as it is a connected arc-transitive normal circulant.

  • (V)

    For any connected arc-transitive circulant Γ, the full automorphism group Aut(Γ) is explicitly given by part (5) of Theorem 1.1. However, it is a challenging problem to identify arc-transitive subgroups of Aut(Γ), which is equivalent to characterizing certain permutation groups that contain a regular cyclic subgroup; refer to [19].

  • (VI)

    Ignoring the orientations of arcs of the circulants, Theorem 1.1 gives a characterization of edge-transitive undirected circulants.

  • (VII)

    The existence of tensor-lexicographic decomposition for connected arc-transitive circulants was obtained in [13, Theorem 4] by the approach of Schur ring, while the uniqueness was not considered.

Theorem 1.1 provides an essential tool to study finite arc-transitive circulants. In many situations, the tensor-lexicographic decomposition in Theorem 1.1 reduces a problem on arc-transitive circulants to the problem on normal ones. We shall illustrate this by verifying the following Theorem 1.3 on the CI-property of arc-transitive circulants, which was claimed in [16, Section 7.3] but did not have any published proof in the literature (to the best of the authors' knowledge).

Given a group G and a nonempty subset S of G, the Cayley digraph of G with connection set S, denoted by Cay(G,S), is the digraph with vertex set G such that a vertex x points to a vertex y if and only if yx1S. It is well known that a digraph Γ=(V,A) is isomorphic to a Cayley digraph of a group G if and only if Aut(Γ) contains a subgroup that is regular on V and isomorphic to G. Thus a circulant is precisely a Cayley digraph of a finite cyclic group up to isomorphism. A Cayley digraph Cay(G,S) is said to be a CI-digraph (with respect to G) if for each subset T of G with Cay(G,S)Cay(G,T) there exists αAut(G) such that T=Sα. A Cayley digraph of a cyclic group G that is a CI-digraph with respect to G is called a CI-circulant.

Theorem 1.3

Every finite connected arc-transitive circulant is a CI-circulant.

The structure of this paper is as follows. In Section 2 we prove the existence of tensor-lexicographic decompositions as described in Theorem 1.1. This is then used, together with some result established in Section 3, to give a proof of Theorem 1.3 in Section 4. Finally, in Section 5, we prove the uniqueness of tensor-lexicographic decompositions, thus completing the proof of Theorem 1.1.

Section snippets

Tensor-lexicographic decompositions

For digraphs Γ=(V1,A1) and Σ=(V2,A2), the tensor product (direct product) Γ×Σ is the digraph with vertex set V1×V2 such that (u1,u2) points to (v1,v2) if and only if (u1,v1)A1 and (u2,v2)A2; the lexicographic product Γ[Σ] is the digraph with vertex set V1×V2 such that (u1,u2) points to (v1,v2) if and only if either (u1,v1)A1, or u1=v1 and (u2,v2)A2. The following lemma follows immediately from the definition of tensor product of digraphs.

Lemma 2.1

Let Γ and Σ be digraphs. Then Aut(Γ×Σ)Aut(Γ)×Aut(Σ),

Normal circulants

Let G be a group. For gG denote by gˆ the permutation of G such that xgˆ=xg for each xG. Then Gˆ:={gˆ|gG} is a regular permutation group on G. For a subset S of G, letAut(G,S)={αAut(G)|Sα=S}. Then GˆAut(G,S) is a subgroup of NAut(Γ)(Gˆ), the normalizer of Gˆ in Aut(Γ), where Γ=Cay(G,S). In fact, we have the following:

Lemma 3.1

([10], [22]) Let Γ=Cay(G,S) be a Cayley digraph of a finite group G. Then NAut(Γ)(Gˆ)=GˆAut(G,S).

By Lemma 3.1 we see that if Γ=Cay(G,S) is a connected arc-transitive normal

CI-property of arc-transitive circulants

In this section we prove that every finite connected arc-transitive circulant is a CI-digraph, thus verifying Theorem 1.3.

Lemma 4.1

([3]) Let Γ be a Cayley digraph of a group G. Then Cay(G,S) is a CI-digraph if and only if each subgroup of Aut(Γ) conjugate to Gˆ in Sym(G) is conjugate to Gˆ in Aut(Γ).

The following lemma is also needed in this section.

Lemma 4.2

Let G be a cyclic transitive subgroup of Sym(Ω)×Sym(Δ) acting on Ω×Δ by product action. Then there exist regular cyclic subgroups H and K of Sym(Ω) and Sym(Δ

Proof of Theorem 1.1

Definition 5.1

A digraph Γ is said to be R-thick if it has distinct vertices u and v such that Γ+(u)=Γ+(v) and Γ(u)=Γ(v). A digraph that is not R-thick is said to be R-thin.

Lemma 5.2

Let Γ be an R-thick vertex-transitive digraph with no loop. Then the following hold:

  • (a)

    ΓΣ[Kb] for some digraph Σ and b2;

  • (b)

    if Γ is a normal Cayley digraph, then Γ has order at most 4;

  • (c)

    if Γ is a normal circulant with nonempty arc set, then ΓC4.

Proof

Let Γ=(V,A), and let G=Aut(Γ). For vV, letB(v)={uV|Γ+(u)=Γ+(v),Γ(u)=Γ(v)}. Then the sets B(v)

Acknowledgements

This project was initiated during the first named author's visit to the University of Melbourne, and was partially supported by National Natural Science Foundation of China (NNSFC 11771200 and 11931005). The authors would like to thank the anonymous referee for careful reading and valuable suggestions on this paper.

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