次单群和特征单群小度数Cayley图的若干性质
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摘要
在群与图的研究中，图的对称性一直是一个热门问题．它主要通过图的自同构群具有某些传递性来描述．这类图的典型代表是Cayley图和Sabidussi陪集图．由于单群是构成群的基石，且具有群的内在性质，因此一直是群论研究的核心．李才恒和徐尚进在文献[1，2]里，证明了有限非交换单群的连通3度弧传递图除两个47!/2阶的非正规5-弧传递图外都是正规的，这个结果不仅圆满解决了有限非交换单群的连通3度弧传递Cayley图的正规性问题，也等于给出了有限非交换单群的连通3度弧传递非正规Cayley图的完全分类．而非可解的次单群是不可解的且只有一个正规子群的有限群，特征单群是同构单群的直积，它们都与单群有着密切的联系．因此研究它们的小度数Cayley图的若干性质是很自然的也是很有意义的．
关于Cayley图对称性的研究关键取决于对其全自同构群了解的深度．众所周知，决定图的自同构群是个基本的问题也是一个很难的问题，只对某些特殊群的Cayley图它们的自同构才被决定，同时Cayley图的正规性也是这方面的一个基本问题．本文围绕这些问题重点考查了非可解次单群中典型的一类群二维线性群的小度数Cayley图的图同构和它们的正规性．
关于Cayley图的同构问题自1967年Adman提出了一个关于每个有限循环群都是DCI-群的猜想以来，一直是该领域的一个非常活跃的问题，李才恒证明了单群都是3-CI群，相应的我们就想知道非可解的次单群是不是3-CI群呢?就这个问题我证明了最小的非可解次单群S_5是非3-CI群．
本文主要围绕以下几个方面展开：
(1)研究了二维线性群PGL(2，p)(p是一个素数)的连通3度和4度边传递Cayley图的自同构群的结构和它们的正规性．
(2)考查了S_5的连通3度Cayley图，给出了它们的完全分类，并得出了它的3度Cayley图都是正规的且是非3-CI的．
(3)研究了非可解特征单群的连通3度弧传递Cayley图的正规性，得出仅当G≤S_(24)且关于它的Cayley图是5-弧传递时才是非正规的．
本文主要采用的是群论方法．文中有关群论和代数图论的概念可参考文献[3，4，5]．
The symmetry of graphs has always been a hot issue in studying group and graph. It is mainly described by some transitivity possessed by the automorphism groups of the graphs. The classical representatives of these graphs are Cayley graph and Sabidussi coset graph. Since simple groups are the base to consititute finite group, and they have the inner prosperity of group, studying the simple group has been a very hot issue problem. In [1, 2], Caiheng Li and Shangjin Xu proved there are only two nonnormal connected cubic arc-transitive Cayley graphs of nonabelian simple group, which two graphs are 5-arc-transitive and their order are (47!)/2. These results not only determine the normality of connected cubic arc-transitive Cayley graph on nonabelian simple group, but also give the compete classical of connected cubic arc-transitive Cayley graph of nonabelian simple group. Nonsolvable sub-simples group is a nonsolvable finite group with only a normal subgroup, and character simple group is a product of isomorphic simple groups, both of them have close relation with simple group. So, it is very natural and interesting to research their small valency Cayley graphs.
The symmetry of Cayley graphs depends on the information how deeply we know from their full automorphism groups. As we know, it is very difficult to determine the automorphism group of graphs, and it is a fundamental problem. Till now, only a few of kinds of group with some characters are known. In additon, the normality is also a fundamental problem for the symmetry of Cayley graphs. In this thesis, about these porblems, we study the autmorphism groups and normality of small valency Cayley graphs on 2-dimensional linear groups which they are classical representatives of nonsolvable sub-simple groups.
Investigating fnite CI-graphs has been a currently very active topic in the area of Cayley graphs since Adam put forward a conjecture which each cyclic group is a CI-group in 1967. Caiheng Li proved that all simple groups are 3-CI groups, accordingly, we want to know whether nonsolvable sub-simple groups are 3-CI groups? About this problem, we prove S_5 which its order is the smallest in nonsovable sub-simple groups is not 3-CI group.
So, in this thesis, my main results are the following items:
QuestionⅠResearch the structure of automorphism groups and normality of connected cubic and tetravelent edge-transitive Cayley graphs on PGL(2,p)(p is a prime).
QuestionⅡDetermine the normality and CI-property of connected cubic Cayley graphs on S_5.
QuestionⅢDiscuss the normality of connected cubic arc-transitive Cayley graphs of nonsolvable character simple groups, and get the Cayley graphs on character simple groups are nonnormal only when G≤S_(24) and their cubic arc-transitive Caylye graphs are 5-arc transitive graphs.
The method used in this thesis is mainly group-theoretic. For concepts of group theory and algebraic graph theory we refer the readers to [3, 4, 5].
关于Cayley图对称性的研究关键取决于对其全自同构群了解的深度．众所周知，决定图的自同构群是个基本的问题也是一个很难的问题，只对某些特殊群的Cayley图它们的自同构才被决定，同时Cayley图的正规性也是这方面的一个基本问题．本文围绕这些问题重点考查了非可解次单群中典型的一类群二维线性群的小度数Cayley图的图同构和它们的正规性．
关于Cayley图的同构问题自1967年Adman提出了一个关于每个有限循环群都是DCI-群的猜想以来，一直是该领域的一个非常活跃的问题，李才恒证明了单群都是3-CI群，相应的我们就想知道非可解的次单群是不是3-CI群呢?就这个问题我证明了最小的非可解次单群S_5是非3-CI群．
本文主要围绕以下几个方面展开：
(1)研究了二维线性群PGL(2，p)(p是一个素数)的连通3度和4度边传递Cayley图的自同构群的结构和它们的正规性．
(2)考查了S_5的连通3度Cayley图，给出了它们的完全分类，并得出了它的3度Cayley图都是正规的且是非3-CI的．
(3)研究了非可解特征单群的连通3度弧传递Cayley图的正规性，得出仅当G≤S_(24)且关于它的Cayley图是5-弧传递时才是非正规的．
本文主要采用的是群论方法．文中有关群论和代数图论的概念可参考文献[3，4，5]．
The symmetry of graphs has always been a hot issue in studying group and graph. It is mainly described by some transitivity possessed by the automorphism groups of the graphs. The classical representatives of these graphs are Cayley graph and Sabidussi coset graph. Since simple groups are the base to consititute finite group, and they have the inner prosperity of group, studying the simple group has been a very hot issue problem. In [1, 2], Caiheng Li and Shangjin Xu proved there are only two nonnormal connected cubic arc-transitive Cayley graphs of nonabelian simple group, which two graphs are 5-arc-transitive and their order are (47!)/2. These results not only determine the normality of connected cubic arc-transitive Cayley graph on nonabelian simple group, but also give the compete classical of connected cubic arc-transitive Cayley graph of nonabelian simple group. Nonsolvable sub-simples group is a nonsolvable finite group with only a normal subgroup, and character simple group is a product of isomorphic simple groups, both of them have close relation with simple group. So, it is very natural and interesting to research their small valency Cayley graphs.
The symmetry of Cayley graphs depends on the information how deeply we know from their full automorphism groups. As we know, it is very difficult to determine the automorphism group of graphs, and it is a fundamental problem. Till now, only a few of kinds of group with some characters are known. In additon, the normality is also a fundamental problem for the symmetry of Cayley graphs. In this thesis, about these porblems, we study the autmorphism groups and normality of small valency Cayley graphs on 2-dimensional linear groups which they are classical representatives of nonsolvable sub-simple groups.
Investigating fnite CI-graphs has been a currently very active topic in the area of Cayley graphs since Adam put forward a conjecture which each cyclic group is a CI-group in 1967. Caiheng Li proved that all simple groups are 3-CI groups, accordingly, we want to know whether nonsolvable sub-simple groups are 3-CI groups? About this problem, we prove S_5 which its order is the smallest in nonsovable sub-simple groups is not 3-CI group.
So, in this thesis, my main results are the following items:
QuestionⅠResearch the structure of automorphism groups and normality of connected cubic and tetravelent edge-transitive Cayley graphs on PGL(2,p)(p is a prime).
QuestionⅡDetermine the normality and CI-property of connected cubic Cayley graphs on S_5.
QuestionⅢDiscuss the normality of connected cubic arc-transitive Cayley graphs of nonsolvable character simple groups, and get the Cayley graphs on character simple groups are nonnormal only when G≤S_(24) and their cubic arc-transitive Caylye graphs are 5-arc transitive graphs.
The method used in this thesis is mainly group-theoretic. For concepts of group theory and algebraic graph theory we refer the readers to [3, 4, 5].
引文
[1] C. H. Li, Isomorphisms of finite Cayley graphs, Dissertation, The University of Western Australia, (1996)
[2] Shangjin Xu, On Graphs of Small Valencies Admitting Transitive Action of Simple Groups, Dissertation, Peking University, (2003).
[3] Ming Yao Xu, the Introduction of Finite Groups, Beijing, the science publisher, (1999).
[4] N.Biggs. Algebraic Graph Theory(Second Edition),Cambridge University Press (1993).
[5] M. Suzuki, Group Theory I, New York, Berlin, Springer, (1986).
[6] J. D. Dixon and B. Motimer, Permutation Groups, Springer, New York, Hong Kong, 1996.
[7] B. Alspach, Isomorphisms of Cayley graphs on abelian groups, in: Graph Symmetry, Algebraic Methods and Applications, NATO ASI Ser. C, (1997), 497, 1-23.
[8] P. P. P'alfy, Isomorphism problem for relational structures with a cyclic automorphism, Eur. J. Combin, (1987), 8, 35-43.
[9] C. E. Praeger, Finite transitive permutation groups and finite vertex-transitive graphs, in Graph Symmetry: Algebraic Methods and Applications, NATO ASI Ser. C, (1997), 497, 277-318.
[10] M. Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math., (1998), 182, 309-320.
[11] C. D. Godsil, On the full automorphism group of a graph, Combinatorica, (1981), 1, 243-256.
[12] W. J. Wong, Determination of a class of primitive permutation groups, Math. Z., (1967), 99, 235-246.
[13] R. Weiss, The nonexistence of 8-transitive graphs, Combinatorica, (1981), 1, 309-311.
[14] C. H. Li, Z. P. Lu and H. Zhang, Tetravalent edge-transitive Cayley graphs with odd number of vertices, J. Combin. Theroy, Ser. B, (2006), 96, 164-181.
[15] X. G. Fang, C. H. Li and M. Y. Xu, On finite edge-transitive Cayley graphs of valency 4, European J. Combin(2004), 25, 1107-1116.
[16]黄本文，一类2~3p~2(p=3，7)阶群的构造，数学杂志， (2000)，1．
[17] Xubo Zhao, The Normality of The Connected Cubic Coset Graphs on Finite groups, Dissertation, The University of Guangxi, (2006).
[18] X. G. Fang, C. H. Li and C. E. Praeger, The locally 2-arc transitive graphs admitting a Ree simple group, J. Algebra, (2004), 282, 638-666.
[19] R. Guralnick, Subgroups of prime power index in a simple group, J. Algebra, (1983), 81, 304-311.
[20] J.Wang, On the maximal subgroups of index 2~a3~b with a, b ≤ 6 in an almost simple group, (preprint) (1998).
[21] N.Biggs and A.T.White, Permutation Groups and Combinatorial Structure, Cambridge University Press (1979).
[22] Z. P. Lu and M. Y. Xu, On the normality of Cayley graphs of order pq, Australas. J. Combin., (2003), 27, 81-93.
[23] Ming Yao Xu, Qin Hai Zhang and Jin Xin Zhou, On the normality of Cayley digraph on abelian group, J. Sys. Sci and Math. Sci, (2005), 25(6), 700-710.
[24] S. J. Xu, X. G. Fang, M. Y. Xu, On cubic s-arc transitive Cayley graphs of finite simple group, European J. Combin, (2005),26, 133-143.
[25] S. J. Xu, X. G. Fang, J. Wang and M. Y. Xu, 5-arc transitive cubic Cayley graphs on finite simple groups, European J. Combin, (2007), 28, 1023-1036.
[26] X. G. Fang, C. H. Li, J. Wang and M. Y. Xu, On cubic normal Cayley graphs of finite simple groups, Discrete Math, (2002), 244, 67-75.
[27] C. E. Praeger, Finite normal edge-transitive Cayley graphs, Bull. Austral. Math. Soc,(1999), 60 , 207-220.
[28] C. H. Li, Finite s-arc transitive Cayley graphs and flag-transitive projective planes, Proc. Amer. Math. Soc, (2005), 133, 31-41.
[29] C. H. Li, Finite edge-transitive Cayley graphs and rotary Cayley maps, Trans. Amer. Math. Soc, (2006), 358, 4605-4635.
[30] A. Adam, Research problem 2-10, J. Combin. Theory 2 (1967) 393.
[31] D. Z. Djokovic, Isomorphism problem for a special class of graphs, Acta Math. Acad. Sci. Hungar. (1970), 21, 267-270.
[32] B. Elspas and J. Turner, Graphs with circulant adjacency matrices, J. Combin. Theory 9 (1970), 297-307.
[33] C.H. Li, On Cayley Isomorphisms of fnite Cayley graphs-a survey, Discrete Mathematics, (2002), 256, 301-334.
[34] C.H. Li, On isomorphisms of connected Cayley graphs II, J. Combin. Theory Ser.B , (1998), 74, 28-34.
[35] Xian Hua Li and Ming Yao Xu, The primitive permutation groups of degree 2~a3~b, Arch. Math, (2006), 86, 385-391.
[36] P. B. Klendman and M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Notes (Cambridge University Press, New York, Sydney, 1990).
[37] C. H. Li, On isomoporphisms of connected Cayley graphs III, Bull, Austral. Math. Soc, (1998), 58, 137-145.
[38]王长群，王殿军，徐明耀，有限群的正规Cayley图，中国科学A辑，(1998)， 28卷第2期，131-139．
[39] Cheryl E. Prager, Quotients and inclusions of finite quasiprimitive permutation groups. Journal of Algebra, (2003), 269 329-346.
[40] M.W. Liebeck, C.E. Praeger, J. Saxl, On the O'Nan-Scott theorem for finite primitive permutation groups, J. Austral. Math. Soc. Ser. (A), (1988), 44(3), 389-396.
[41] C.E. Praeger, An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc, (1993), 47(2), 227-239.
[42] Robet W.Basseley, Cheryl E. Praeger, Csaba Schneider, Quasiprimitive groups and blowupdecompositipns, Journal of Algebra, 2006, 1-15.
[43] C.E. Praeger, Finite quasiprimitive graphs, in: R.A. Bailey (Ed.), Surveys in Combinatorics,1997, London, Cambridge University Press, Cambridge, 1997, 65-85.
[44] Baddeley, Robert W, Praeger, Cheryl E. On primitive overgroups of quasiprimitive permutationgroups. J. Algebra, (2003), 263(2), 294-344.
[2] Shangjin Xu, On Graphs of Small Valencies Admitting Transitive Action of Simple Groups, Dissertation, Peking University, (2003).
[3] Ming Yao Xu, the Introduction of Finite Groups, Beijing, the science publisher, (1999).
[4] N.Biggs. Algebraic Graph Theory(Second Edition),Cambridge University Press (1993).
[5] M. Suzuki, Group Theory I, New York, Berlin, Springer, (1986).
[6] J. D. Dixon and B. Motimer, Permutation Groups, Springer, New York, Hong Kong, 1996.
[7] B. Alspach, Isomorphisms of Cayley graphs on abelian groups, in: Graph Symmetry, Algebraic Methods and Applications, NATO ASI Ser. C, (1997), 497, 1-23.
[8] P. P. P'alfy, Isomorphism problem for relational structures with a cyclic automorphism, Eur. J. Combin, (1987), 8, 35-43.
[9] C. E. Praeger, Finite transitive permutation groups and finite vertex-transitive graphs, in Graph Symmetry: Algebraic Methods and Applications, NATO ASI Ser. C, (1997), 497, 277-318.
[10] M. Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math., (1998), 182, 309-320.
[11] C. D. Godsil, On the full automorphism group of a graph, Combinatorica, (1981), 1, 243-256.
[12] W. J. Wong, Determination of a class of primitive permutation groups, Math. Z., (1967), 99, 235-246.
[13] R. Weiss, The nonexistence of 8-transitive graphs, Combinatorica, (1981), 1, 309-311.
[14] C. H. Li, Z. P. Lu and H. Zhang, Tetravalent edge-transitive Cayley graphs with odd number of vertices, J. Combin. Theroy, Ser. B, (2006), 96, 164-181.
[15] X. G. Fang, C. H. Li and M. Y. Xu, On finite edge-transitive Cayley graphs of valency 4, European J. Combin(2004), 25, 1107-1116.
[16]黄本文，一类2~3p~2(p=3，7)阶群的构造，数学杂志， (2000)，1．
[17] Xubo Zhao, The Normality of The Connected Cubic Coset Graphs on Finite groups, Dissertation, The University of Guangxi, (2006).
[18] X. G. Fang, C. H. Li and C. E. Praeger, The locally 2-arc transitive graphs admitting a Ree simple group, J. Algebra, (2004), 282, 638-666.
[19] R. Guralnick, Subgroups of prime power index in a simple group, J. Algebra, (1983), 81, 304-311.
[20] J.Wang, On the maximal subgroups of index 2~a3~b with a, b ≤ 6 in an almost simple group, (preprint) (1998).
[21] N.Biggs and A.T.White, Permutation Groups and Combinatorial Structure, Cambridge University Press (1979).
[22] Z. P. Lu and M. Y. Xu, On the normality of Cayley graphs of order pq, Australas. J. Combin., (2003), 27, 81-93.
[23] Ming Yao Xu, Qin Hai Zhang and Jin Xin Zhou, On the normality of Cayley digraph on abelian group, J. Sys. Sci and Math. Sci, (2005), 25(6), 700-710.
[24] S. J. Xu, X. G. Fang, M. Y. Xu, On cubic s-arc transitive Cayley graphs of finite simple group, European J. Combin, (2005),26, 133-143.
[25] S. J. Xu, X. G. Fang, J. Wang and M. Y. Xu, 5-arc transitive cubic Cayley graphs on finite simple groups, European J. Combin, (2007), 28, 1023-1036.
[26] X. G. Fang, C. H. Li, J. Wang and M. Y. Xu, On cubic normal Cayley graphs of finite simple groups, Discrete Math, (2002), 244, 67-75.
[27] C. E. Praeger, Finite normal edge-transitive Cayley graphs, Bull. Austral. Math. Soc,(1999), 60 , 207-220.
[28] C. H. Li, Finite s-arc transitive Cayley graphs and flag-transitive projective planes, Proc. Amer. Math. Soc, (2005), 133, 31-41.
[29] C. H. Li, Finite edge-transitive Cayley graphs and rotary Cayley maps, Trans. Amer. Math. Soc, (2006), 358, 4605-4635.
[30] A. Adam, Research problem 2-10, J. Combin. Theory 2 (1967) 393.
[31] D. Z. Djokovic, Isomorphism problem for a special class of graphs, Acta Math. Acad. Sci. Hungar. (1970), 21, 267-270.
[32] B. Elspas and J. Turner, Graphs with circulant adjacency matrices, J. Combin. Theory 9 (1970), 297-307.
[33] C.H. Li, On Cayley Isomorphisms of fnite Cayley graphs-a survey, Discrete Mathematics, (2002), 256, 301-334.
[34] C.H. Li, On isomorphisms of connected Cayley graphs II, J. Combin. Theory Ser.B , (1998), 74, 28-34.
[35] Xian Hua Li and Ming Yao Xu, The primitive permutation groups of degree 2~a3~b, Arch. Math, (2006), 86, 385-391.
[36] P. B. Klendman and M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Notes (Cambridge University Press, New York, Sydney, 1990).
[37] C. H. Li, On isomoporphisms of connected Cayley graphs III, Bull, Austral. Math. Soc, (1998), 58, 137-145.
[38]王长群，王殿军，徐明耀，有限群的正规Cayley图，中国科学A辑，(1998)， 28卷第2期，131-139．
[39] Cheryl E. Prager, Quotients and inclusions of finite quasiprimitive permutation groups. Journal of Algebra, (2003), 269 329-346.
[40] M.W. Liebeck, C.E. Praeger, J. Saxl, On the O'Nan-Scott theorem for finite primitive permutation groups, J. Austral. Math. Soc. Ser. (A), (1988), 44(3), 389-396.
[41] C.E. Praeger, An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc, (1993), 47(2), 227-239.
[42] Robet W.Basseley, Cheryl E. Praeger, Csaba Schneider, Quasiprimitive groups and blowupdecompositipns, Journal of Algebra, 2006, 1-15.
[43] C.E. Praeger, Finite quasiprimitive graphs, in: R.A. Bailey (Ed.), Surveys in Combinatorics,1997, London, Cambridge University Press, Cambridge, 1997, 65-85.
[44] Baddeley, Robert W, Praeger, Cheryl E. On primitive overgroups of quasiprimitive permutationgroups. J. Algebra, (2003), 263(2), 294-344.