一类Cayley图的自同构群
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摘要
给定有限群G和它的一个满足S=S-1={s-1|s∈S}和1(?)S的子集S.群G关于S的Cayley图Cay(G,S)定义为具有顶点集G和边集{{g,h}|g,h∈G,gh-1∈S}的图.给定g∈G,定义R(g):h(?)hg,(?)h∈G.映射R:g(?)R(g),(?)g∈G为一个同态映射,其同态像称为群G的右正则表示,记为R(G).易证,R(G)为Aut(Cay(G,S))的正则子群.若R(G)在Aut(Cay(G,S))中正规,则称Cayley图Cay(G,S)为正规的.
设n为正整数,Cn[2K1]表示长为n的圈Cn和两点空图2K1的字典式积,即顶点集合为{xi,yi|i∈Zn},边集合为{{xi,xi+1},{yi,yi+1},{xi,yi+1},{yi,xi+1}|i∈Zn}的图.
本文主要围绕Cn[2K1]的自同构群展开研究.得到了如下结论：(1)当n=4时,Aut(Cn[2K1])≌S42(?)Z2,当n>4时,Z2n(?)D2n;(2)决定了所有群G及其Cayley子集S使得Cn[2K1]≌Cay(G,S).
Let G be a finite group and S a subset of G such that S=S-1= {s-1|s∈S)and 1(?)S.The Cayley graph Cay(G,S)on G with respect to S is defined with vertex set G and edge set{(g,sg)|g∈G,s∈S}.For g∈G, R(g)is defined by:h(?)hg,(?)h∈G.The map R:9(?)R(g),(?)g∈G is an homomorphism,and its homomorphic image is the right regular representation of G,denote by R(G).Obviously, R(G)is a regular subgroup of Aut(Cay(G,S)).A Cayley graph Cay(G,S)is called normal if the right regular representation R(G) of G is a normal subgroup of Aut(Cay(G,s)).
Let n be a positive integer.The graph Cn[2K1]is a lexicographic product of Cn and 2K1 with vertex set{xi,yi|i∈Zn}and edge set{{xi,xi+1},{yi,yi+1},{xi, yi+1},{yi,xi+1}|i∈Zn}.
The work in this paper is mainly about the automorphism group of Cn[2K1]. We have the following conclusions:(1)Aut(Cn[2K1])is isomorphic to S42(?)Z2 for n=4,or to Z2n(?)D2n for n>4；(2)we have determined all pairs of group G and Cayley subset S of G satisfying Cn[2K1]≌Cay(G,S).
设n为正整数,Cn[2K1]表示长为n的圈Cn和两点空图2K1的字典式积,即顶点集合为{xi,yi|i∈Zn},边集合为{{xi,xi+1},{yi,yi+1},{xi,yi+1},{yi,xi+1}|i∈Zn}的图.
本文主要围绕Cn[2K1]的自同构群展开研究.得到了如下结论：(1)当n=4时,Aut(Cn[2K1])≌S42(?)Z2,当n>4时,Z2n(?)D2n;(2)决定了所有群G及其Cayley子集S使得Cn[2K1]≌Cay(G,S).
Let G be a finite group and S a subset of G such that S=S-1= {s-1|s∈S)and 1(?)S.The Cayley graph Cay(G,S)on G with respect to S is defined with vertex set G and edge set{(g,sg)|g∈G,s∈S}.For g∈G, R(g)is defined by:h(?)hg,(?)h∈G.The map R:9(?)R(g),(?)g∈G is an homomorphism,and its homomorphic image is the right regular representation of G,denote by R(G).Obviously, R(G)is a regular subgroup of Aut(Cay(G,S)).A Cayley graph Cay(G,S)is called normal if the right regular representation R(G) of G is a normal subgroup of Aut(Cay(G,s)).
Let n be a positive integer.The graph Cn[2K1]is a lexicographic product of Cn and 2K1 with vertex set{xi,yi|i∈Zn}and edge set{{xi,xi+1},{yi,yi+1},{xi, yi+1},{yi,xi+1}|i∈Zn}.
The work in this paper is mainly about the automorphism group of Cn[2K1]. We have the following conclusions:(1)Aut(Cn[2K1])is isomorphic to S42(?)Z2 for n=4,or to Z2n(?)D2n for n>4；(2)we have determined all pairs of group G and Cayley subset S of G satisfying Cn[2K1]≌Cay(G,S).
引文
[1]B. Alspach, Point-symmetric graphs and digraphs of prime order and transitive permutation groups of prime degree, Combin. Theory,15 (1973) 12-17.
[2]Y.G. Baik, Y.Q. Feng, H.S. Sim, M.Y. Xu, On the normality of Cayley graph of abelian groups, Algebra Colloq.,5 (1998) 297-304.
[3]Y.G. Baik, Y.Q. Feng, H.S. Sim, The normality of Cayley graphs of finite Abelian groups with valency 5, Systems Science and Mathematical Sciences,13 (2000) 425-431.
[4]E. Dobson, D. Witte, Transitive premutation groups of prime-squared degree, J. Algebraic Combin.,16 (2002) 43-69.
[5]S.F. Du, R.J. Wang, M.Y. Xu, On the normality of Cayley digraph of group of order twice a prime, Austral. J. Combin.,118 (1998) 227-234.
[6]X.G. Fang, C.H. Li, D. J. Wang, M.Y. Xu, On cubic Cayley graphs of finite simple groups, Discrete Math.,244 (2002) 67-75.
[7]Y.Q. Feng, D.J. Wang, J.L. Chen, A family of non-normal Cayley digraphs, Acta. Math. Sinica. New series,17 (2001) 147-152.
[8]Y.Q. Feng, J.H. Kwak, R.J. Wang, Automorphism Groups of 4-valent Connected Cayley Graphs of p-Groups, Chin. Ann. of Math.,3(2001) 281-286.
[9]Y.Q. Feng, M.Y. Xu, Automorphism groups of tetravalent Cayley graphs on regular p-groups, Discrete Math.,305 (2005) 354-360.
[10]I. Kovacs, B. Kuzamn, A. Malnic, On non-normal arc transitive 4-valent dihedrants, Acta Mathematica Sinica English Series,6 (2009) 1-15.
[11]Z.P. Lu, M.Y. Xu, On the normality of Cayley graphs of order pq, Austral. J. Combin.,27 (2003) 81-93.
[12]聂淑帆,冯衍全,4p阶群上2度Cayley有向图的正规性,北京交通大学学报,3(2006)81-83.
[13]C.Q. Wang, M.Y. Xu, Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D2n, European J. Combin.,27 (2006) 750-766.
[14]C.Q. Wang, D.J. Wang, M. Y. Xu, On normal Cayley graphs of finite groups, Science in China A,28 (1998) 131-139.
[15]S.J. Xu, X.G. Fang, J. Wang, M.Y. Xu, On cubic s-arc transitive Cayley graphs of finite simple groups, European J. Combin.,26 (2005) 133-143.
[16]S.J. Xu, X.G. Fang, J. Wang, M.Y. Xu,5-Arc transitive cubic Cayley graphs of finite simple groups, European J. Combin.,28 (2007) 1023-1036.
[17]M.Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math.,182 (1998) 309-319.
[18]徐明曜,张勤海，周进鑫,关于交换群上的Cayley有向图的正规性,系统科学与数学,25(2005)700-710.
[19]C. Zhang, J.X. Zhou, Y.Q. Feng, Automorphisms of cubic Cayley graphs of order 2pq, Discrete Math.,309 (2009) 2687-2695.
[20]J. Zhou, The normality of Cayley digraphs of out-valency 3 of dihedral group of order 4p, J. Northeast. Math.,15 (1999) 173-182.
[21]J.X. Zhou, Tetravalent non-normal Cayley Graphs of order 4p, Electornic J. Combin.,16 (2009) R118.
[2]Y.G. Baik, Y.Q. Feng, H.S. Sim, M.Y. Xu, On the normality of Cayley graph of abelian groups, Algebra Colloq.,5 (1998) 297-304.
[3]Y.G. Baik, Y.Q. Feng, H.S. Sim, The normality of Cayley graphs of finite Abelian groups with valency 5, Systems Science and Mathematical Sciences,13 (2000) 425-431.
[4]E. Dobson, D. Witte, Transitive premutation groups of prime-squared degree, J. Algebraic Combin.,16 (2002) 43-69.
[5]S.F. Du, R.J. Wang, M.Y. Xu, On the normality of Cayley digraph of group of order twice a prime, Austral. J. Combin.,118 (1998) 227-234.
[6]X.G. Fang, C.H. Li, D. J. Wang, M.Y. Xu, On cubic Cayley graphs of finite simple groups, Discrete Math.,244 (2002) 67-75.
[7]Y.Q. Feng, D.J. Wang, J.L. Chen, A family of non-normal Cayley digraphs, Acta. Math. Sinica. New series,17 (2001) 147-152.
[8]Y.Q. Feng, J.H. Kwak, R.J. Wang, Automorphism Groups of 4-valent Connected Cayley Graphs of p-Groups, Chin. Ann. of Math.,3(2001) 281-286.
[9]Y.Q. Feng, M.Y. Xu, Automorphism groups of tetravalent Cayley graphs on regular p-groups, Discrete Math.,305 (2005) 354-360.
[10]I. Kovacs, B. Kuzamn, A. Malnic, On non-normal arc transitive 4-valent dihedrants, Acta Mathematica Sinica English Series,6 (2009) 1-15.
[11]Z.P. Lu, M.Y. Xu, On the normality of Cayley graphs of order pq, Austral. J. Combin.,27 (2003) 81-93.
[12]聂淑帆,冯衍全,4p阶群上2度Cayley有向图的正规性,北京交通大学学报,3(2006)81-83.
[13]C.Q. Wang, M.Y. Xu, Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D2n, European J. Combin.,27 (2006) 750-766.
[14]C.Q. Wang, D.J. Wang, M. Y. Xu, On normal Cayley graphs of finite groups, Science in China A,28 (1998) 131-139.
[15]S.J. Xu, X.G. Fang, J. Wang, M.Y. Xu, On cubic s-arc transitive Cayley graphs of finite simple groups, European J. Combin.,26 (2005) 133-143.
[16]S.J. Xu, X.G. Fang, J. Wang, M.Y. Xu,5-Arc transitive cubic Cayley graphs of finite simple groups, European J. Combin.,28 (2007) 1023-1036.
[17]M.Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math.,182 (1998) 309-319.
[18]徐明曜,张勤海，周进鑫,关于交换群上的Cayley有向图的正规性,系统科学与数学,25(2005)700-710.
[19]C. Zhang, J.X. Zhou, Y.Q. Feng, Automorphisms of cubic Cayley graphs of order 2pq, Discrete Math.,309 (2009) 2687-2695.
[20]J. Zhou, The normality of Cayley digraphs of out-valency 3 of dihedral group of order 4p, J. Northeast. Math.,15 (1999) 173-182.
[21]J.X. Zhou, Tetravalent non-normal Cayley Graphs of order 4p, Electornic J. Combin.,16 (2009) R118.