Automorphism groups of Cayley digraphs
Automorphism groups of Cayley digraphs
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DOI：
10.1201/9780203885765-5
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发表时间：
2008-07
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通讯作者：
Yan-Quan Feng;Z.-P. Lu M.-Y. Xu
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文献类型：
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作者：
Yan-Quan Feng;Z.-P. Lu M.-Y. Xu
让G是一组G 1�S和S⊂凯莱有向图礁(G, S) G对S有向图的顶点集G, x, y∈G,那里是一个有向边从x到y每当yx受到打击还得追溯到∈f−−1 1 = S,然后礁(G, S)可以被视为一个(无向)图通过识别两个定向边缘(x, y)和x (y)与一个边缘{x, y}。设X = Cay（G， S）是一个Cayley有向图。那么每一个元素g∈g通过将每个顶点X映射到xg自然地推导出X的自同构R(g)。如果R(G) ={R(G) | G∈G}是x的自同构群的正规子群，则称Cayley有向图Cay（G， S）是正规的。本文简要综述了近年来关于Cayley有向图自同构群的研究成果，重点讨论了Cayley有向图的正规性。在本文中，除非另有说明，图或有向图（有向图）是有限和简单的。对于（di）图X，我们分别用V (X)、E(X)和Aut(X)表示X的顶点集、边集和自同构群。如果Aut(X)分别作用于V (X)或E(X)，则称（di）图为顶点传递图或边传递图。注意，对于一个（无向）图X， X的每条边{u， v}给出两个有序对（u, v）和（v, u），称为X的弧。因此，我们有时，如果必要的话，把图X看作一个有向图。设G是G的一个群，S是G的一个子集，使得1′∈S。G上关于S的Cayley有向图Cay（G， S）定义为顶点集G和边集{(G, sg) | G∈G， S∈S}的有向图。对于Cayley有向图X = Cay(G， S)，为方便起见，我们通常称| S |为X的价。如果S是对称的，即S−1 ={S−1 | S∈S} = S，则Cay（G， S）可以看作是一个无向图，通过用一条无向边识别两条对向边。我们有时称Cayley有向图Cay（G， S）为Cayley图，如果S是对称的，则称Cay（G， S）为有向Cayley图，以强调S - 1 S。设X = Cay（G， S）为Cayley有向图。用右乘法考虑G对V (X)的作用。那么每一个元素g∈g通过将每个顶点X映射到xg自然地推导出X的自同构R(g)。设R(G) ={R(G) | G∈G}。则R(G)是Aut(X)和R(G) ~ G的子群，因此X是一个顶点传递有向图。显然，R(G)有规律地作用于顶点，即R(G)在顶点上是可传递的，并且只有R(G)的单位元固定任何给定的顶点。更进一步，我们知道有向图Y与群G上的Cayley有向图同构当且仅当它的自同构群包含一个与G同构的子群，并且正则地作用于Y的顶点（见引理16.3）)。注意到R(G)在V (X)上是正则的，它意味着Aut(X) = R(G)Aut(X)1。
Let G be a group and S ⊂ G with 1 � S. A Cayley digraph Cay(G, S) on G with respect to S is the digraph with vertex set G such that, for x, y ∈ G , there is a directed edge from x to y whenever yx −1 ∈ S.I fS −1 = S, then Cay(G, S) can be viewed as an (undirected) graph by identifying two directed edges (x, y) and ( y, x) with one edge {x, y}. Let X = Cay(G, S) be a Cayley digraph. Then every element g ∈ G induces naturally an automorphism R(g) of X by mapping each vertex x to xg. The Cayley digraph Cay(G, S) is said to be normal if R(G) ={ R(g)|g ∈ G} is a normal subgroup of the automorphism group of X. In this paper we shall give a brief survey of recent results on automorphism groups of Cayley digraphs concentrating on the normality of Cayley digraphs. Throughout this paper graphs or digraphs (directed graphs) are finite and simple unless specified otherwise. For a (di)graph X , we denote by V (X ), E(X ) and Aut(X ) the vertex set, the edge set and the automorphism group of X , respectively. A (di)graph is said to be vertex-transitive or edge- transitive if Aut(X ) acts transitively on V (X ) or E(X ), respectively. Note that for an (undirected) graph X , each edge {u, v} of X gives two ordered pairs (u, v) and (v, u), called arcs of X. Thus we sometimes, if necessary, view a graph X as a digraph. Let G be a group and S a subset of G such that 1 �∈ S. The Cayley digraph Cay(G, S) on G with respect to S is defined as the directed graph with vertex set G and edge set {(g, sg) | g ∈ G, s ∈ S}. For a Cayley digraph X = Cay(G, S), we always call | S | the valency of X for convenience. If S is symmetric, that is, if S −1 ={ s −1 | s ∈ S} is equal to S, then Cay(G, S) can be viewed as an undirected graph by identifying two oppositely directed edges with one undirected edge. We sometimes call a Cayley digraph Cay(G, S) a Cayley graph if S is symmetric, and say Cay(G, S) a directed Cayley graph to emphasize S −1 � S. Let X = Cay(G, S) be a Cayley digraph. Consider the action of G on V (X ) by right multiplica- tion. Then every element g ∈ G induces naturally an automorphism R(g) of X by mapping each vertex x to xg. Set R(G) ={ R( g) | g ∈ G}. Then R(G) is a subgroup of Aut(X ) and R(G) ∼ G. Thus X is a vertex-transitive digraph. Clearly, R(G) acts regularly on vertices, that is, R(G) is transitive on vertices and only the identity element of R(G) fixes any given vertex. Further, it is well-known that a digraph Y is isomorphic to a Cayley digraph on some group G if and only if its automorphism group contains a subgroup isomorphic to G, acting regularly on the vertices of Y (see (5, Lemma 16.3)). Noting that R(G) is regular on V (X ), it implies Aut(X ) = R(G)Aut(X )1.