An explicit characterization of arc-transitive circulants
Introduction
Throughout this paper, a digraph is an ordered pair 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 acts transitively on the arc set. A digraph Γ is called a circulant if has a finite cyclic subgroup that is regular on the vertex set, and is called a normal circulant if 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 , and their lexicographic product is denoted by (see Section 2 for the definitions of these products). For a positive integer n, denote by the symmetric group of degree n, and denote by and the complete graph and the edgeless graph of order n, respectively. Let 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 of order and positive integers , where , such that the following hold: ; for ; , and are pairwise coprime; ; .
Moreover, Γ is uniquely determined by the triple 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 of a finite connected arc-transitive normal circulant , a (not necessarily nonempty) set 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 in the statement of Theorem 1.1 is a single loop if and has no loop if . In the former case, the tensor product of with any digraph is isomorphic to the digraph, and hence the digraph Γ described in part (3) of Theorem 1.1 is isomorphic to .
- (II)
For each arc-transitive circulant of order and positive integers such that are pairwise coprime, the digraph is an arc-transitive circulant by Lemma 2.1, Lemma 2.2. The condition as in (2) of Theorem 1.1 ensures that is a non-normal circulant. If , then since , we deduce from Lemma 2.3, Lemma 2.4 that .
- (III)
To be precise, the statement “Γ is uniquely determined by the triple ” means that if and are two tensor-lexicographic decompositions of Γ then , and . Theorem 1.1 shows that every connected arc-transitive circulant has a unique tensor-lexicographic decomposition.
- (IV)
Further descriptions of 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 is explicitly given by part (5) of Theorem 1.1. However, it is a challenging problem to identify arc-transitive subgroups of , 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 , is the digraph with vertex set G such that a vertex x points to a vertex y if and only if . It is well known that a digraph is isomorphic to a Cayley digraph of a group G if and only if 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 is said to be a CI-digraph (with respect to G) if for each subset T of G with there exists such that . 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 and , the tensor product (direct product) is the digraph with vertex set such that points to if and only if and ; the lexicographic product is the digraph with vertex set such that points to if and only if either , or and . The following lemma follows immediately from the definition of tensor product of digraphs.
Lemma 2.1 Let Γ and Σ be digraphs. Then ,
Normal circulants
Let G be a group. For denote by the permutation of G such that for each . Then is a regular permutation group on G. For a subset S of G, let Then is a subgroup of , the normalizer of in , where . In fact, we have the following:
Lemma 3.1 ([10], [22]) Let be a Cayley digraph of a finite group G. Then .
By Lemma 3.1 we see that if 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 is a CI-digraph if and only if each subgroup of conjugate to in is conjugate to in .
The following lemma is also needed in this section.
Lemma 4.2 Let G be a cyclic transitive subgroup of acting on by product action. Then there exist regular cyclic subgroups H and K of and
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 and . 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: for some digraph Σ and ; if Γ is a normal Cayley digraph, then Γ has order at most 4; if Γ is a normal circulant with nonempty arc set, then .
Proof Let , and let . For , let Then the sets
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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