具有几乎单群弧传递作用的图
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摘要
在图的对称性研究中,确定图的自同构群是具有基本重要性的工作.本文主要研究具有有限几乎单群G-弧传递作用图Γ的全自同构群Aut(Γ).
首先,本文提出了T-正规性的概念,这是Cayley图正规性的自然推广,给出了具有单群T-传递作用的图具有T-正规性的充分条件.利用这一充分条件,证明了度数不大于20或是素数度的具有几乎单群G-弧传递作用的图,除了有限个例外,都具有soc(G)-正规性.特别的,在图的度数为3时,排除了所有的例外.利用T-正规性确定了具有几乎单群弧传递作用图的自同构群结构.
然后,本文给出了一种构造具有几乎单群弧传递作用图的方法,利用这种方法构造了Aut(Γ)-非拟本原但G-拟本原的G-弧传递图Γ的两个无限族.据作者所知,目前这样的图只有两个无限族,是分别由李才恒[53]和方新贵等人[36]构造的.
最后,作为上述结果的应用,本文确定了3度的G-弧传递图Γ的全自同构群Aut(Γ),即证明了Aut(Γ) = Ree(q)或者Aut(Γ) =Ree(q)×Z_2,其中G = Ree(q) (q≥27);并且构造出了所有此类图,其中存在图Γ使得Aut(Γ) = Ree(q)×Z_2.
A fundamental problem in determining the structure of a graphΓisthe problem of finding its full automorphism group Aut(Γ). This work ismainly to investigate the full automorphism group Aut(Γ) ofΓ, given itsalmost simple subgroup G.
First, a new concept, namely, T-normal graph is introduced, whichis a natural generalization of the concept of normal Cayley graph. Thenwe consider T-vertex-transitive graphs with T a nonabelian simple groupand obtain a sufficient condition under which one can guarantee thatΓisa T-normal graph. Applying the result to G-arc-transitive graphsΓ, weprove that if the valency v(Γ) ofΓis at most 20 or a prime, thenΓisa soc(G)-normal graph for all but finite possibilities of soc(G). In par-ticular, if v(Γ) = 3, we eliminate all exceptions, that is, ifΓis a cubicG-arc-transitive graph thenΓis soc(G)-normal. By the T-normality, wedetermine the structure of the automorphism group Aut(Γ) ofΓ.
Next, a construction is given for an infinite family of G-arc-transitivegraphs. Using this construction we give two family of quasiprimitve arc-transitive graphs which have non-quasiprimitive full automorphism groups.To the author’s best knowledge, the only two infinite families of suchgraphs are constructed by Li [53] and Fang et al. [36], respectively.
Finally, as an application of the above results, we completely deter-mine the automorphism group Aut(Γ) of cubic G-arc-transitive graphΓwith G = Ree(q)(q≥27), and, namely, we prove that Aut(Γ) = Ree(q) or Aut(Γ) = Ree(q)×Z_2. Moreover, we construct all cubic Ree(q)-arc-transitive graphs, amongst which exits there graphsΓsuch that Aut(Γ) =Ree(q)×Z_2.
首先,本文提出了T-正规性的概念,这是Cayley图正规性的自然推广,给出了具有单群T-传递作用的图具有T-正规性的充分条件.利用这一充分条件,证明了度数不大于20或是素数度的具有几乎单群G-弧传递作用的图,除了有限个例外,都具有soc(G)-正规性.特别的,在图的度数为3时,排除了所有的例外.利用T-正规性确定了具有几乎单群弧传递作用图的自同构群结构.
然后,本文给出了一种构造具有几乎单群弧传递作用图的方法,利用这种方法构造了Aut(Γ)-非拟本原但G-拟本原的G-弧传递图Γ的两个无限族.据作者所知,目前这样的图只有两个无限族,是分别由李才恒[53]和方新贵等人[36]构造的.
最后,作为上述结果的应用,本文确定了3度的G-弧传递图Γ的全自同构群Aut(Γ),即证明了Aut(Γ) = Ree(q)或者Aut(Γ) =Ree(q)×Z_2,其中G = Ree(q) (q≥27);并且构造出了所有此类图,其中存在图Γ使得Aut(Γ) = Ree(q)×Z_2.
A fundamental problem in determining the structure of a graphΓisthe problem of finding its full automorphism group Aut(Γ). This work ismainly to investigate the full automorphism group Aut(Γ) ofΓ, given itsalmost simple subgroup G.
First, a new concept, namely, T-normal graph is introduced, whichis a natural generalization of the concept of normal Cayley graph. Thenwe consider T-vertex-transitive graphs with T a nonabelian simple groupand obtain a sufficient condition under which one can guarantee thatΓisa T-normal graph. Applying the result to G-arc-transitive graphsΓ, weprove that if the valency v(Γ) ofΓis at most 20 or a prime, thenΓisa soc(G)-normal graph for all but finite possibilities of soc(G). In par-ticular, if v(Γ) = 3, we eliminate all exceptions, that is, ifΓis a cubicG-arc-transitive graph thenΓis soc(G)-normal. By the T-normality, wedetermine the structure of the automorphism group Aut(Γ) ofΓ.
Next, a construction is given for an infinite family of G-arc-transitivegraphs. Using this construction we give two family of quasiprimitve arc-transitive graphs which have non-quasiprimitive full automorphism groups.To the author’s best knowledge, the only two infinite families of suchgraphs are constructed by Li [53] and Fang et al. [36], respectively.
Finally, as an application of the above results, we completely deter-mine the automorphism group Aut(Γ) of cubic G-arc-transitive graphΓwith G = Ree(q)(q≥27), and, namely, we prove that Aut(Γ) = Ree(q) or Aut(Γ) = Ree(q)×Z_2. Moreover, we construct all cubic Ree(q)-arc-transitive graphs, amongst which exits there graphsΓsuch that Aut(Γ) =Ree(q)×Z_2.
引文
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[30] X.G.Fang, Construction and classification of some families of almostsimple 2-arc transitive graphs, Ph.D. Thesis, University of WesternAustralia, 1995.
[31] X.G.Fang, L.J.Jia and J.Wang, On the automorphism groups of sym-metric graphs admitting an almost simple group, to appear in Europ.J. Combinatoriacs.
[32] X.G.Fang J.Wang and M.Y.Xu, On 1-arc-regular graphs, Europ. J.Combinatorics 23 (2002), 785-791.
[33] X.G.Fang, C.H.Li and J.Wang, Finite vertex primitive 2-arc regulargraphs, J. Algebraic Combin. 25 (2007), 125-140.
[34] X.G. Fang, C.H. Li, and M.Y. Xu, On edge-transitive Cayley Graphsof Valency four, Europ. J. Combinatorics 25(2004), 1107-1116.
[35] X.G. Fang, C.H. Li and J. Wang, M.Y. Xu, On cubic Cayley graphsof finite simple groups, Discrete Mathematics, 244(2002), 67-75.
[36] X.G.Fang, G.Havas and J.Wang, A family of non-quasiprimitivegraphs admitting a quasiprimitive 2-arc transitive group action, Eu-rop. J. Combin. (1999)20, 551-557.
[37] Y.Q. Feng and M.Y. Xu, Normality of tetravalent Cayley graphs ofodd prime-cube order and its application, Acta Math. Sin. (Engl.Ser.) 21 (2005), no. 4, 903–912.
[38] The GAP Group, GAP–Groups, Algorithms and Programming, Ver-sion 4.4.
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