32p阶二面体群的3度Cayley图的正规性
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摘要
群G关于其不包含单位元1的子集S的Cayley图Γ∶=Cay(G,S)称为正规的,如果G的右正则表示R(G)在Aut(Γ)中正规;称图Γ是G的正则表示(GRR),如果R(G)=Aut(Γ)且Γ是无向图.该文完全解决了32p阶二面体群G=〈a,b|a16p=b2=1,ab=a-1〉(其中p是奇素数)的连通3度无向Cayley图的正规性问题,并获得了该群的一批3度GRR的例子.
A Cayley graphΓ∶=Cay(G,S)of the group G with its subset S1 is said to be normal if R(G),the group of right multiplications is normal in Aut(Γ);The graphΓis said GRR if R(G)=Aut(Γ)andΓis undirected.In this paper,we solve completely the problem for the normality of3-valent Cayley graphs of dihedral groups with order 32 p,G=〈a,b|a16 p=b2=1,ab=a-1〉,where pis an odd prime.In addition,we obtain many infinite families of 3-valent graphical regular representations for the group G.
A Cayley graphΓ∶=Cay(G,S)of the group G with its subset S1 is said to be normal if R(G),the group of right multiplications is normal in Aut(Γ);The graphΓis said GRR if R(G)=Aut(Γ)andΓis undirected.In this paper,we solve completely the problem for the normality of3-valent Cayley graphs of dihedral groups with order 32 p,G=〈a,b|a16 p=b2=1,ab=a-1〉,where pis an odd prime.In addition,we obtain many infinite families of 3-valent graphical regular representations for the group G.
引文
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[7]毛争艳.12p阶二面体群的3度Cayley图的正规性[D].郑州:郑州大学,2012.
[8]姚卿卿.16p阶二面体群的3度Cayley图的正规性[D].郑州:郑州大学,2014.
[9]徐明曜.有限群导引(上下册)[M].2版.北京:科学出版社,1999.
[2]Du S F,Wang R J,Xu M Y.On the normality of Cayley digraphs of order twice a prime[J].Australiasian J Combin,1998,18:227-234.
[3]Dobson H,Witte D.Transitive permutation groups of prime-squared degree[J].J Algebraic Combin,2002,16:43-69.
[4]Li C H.Isomorphisms of finite Cayley graphs[D].Perth:The Uinversity of Western Australia,1996.
[5]Xu S J,Fang X G,Wang J,et al.On cubic s-arc transitive Cayley graphs of finite simple groups[J].European Journal of Combinatorics,2005,26:133-143.
[6]Xu S J,Fang X G,Wang J,et al.5-arc transitive cubic Cayley graphs on finite simple groups[J].European Journal of Combinatorics,2007,28:1023-1036.
[7]毛争艳.12p阶二面体群的3度Cayley图的正规性[D].郑州:郑州大学,2012.
[8]姚卿卿.16p阶二面体群的3度Cayley图的正规性[D].郑州:郑州大学,2014.
[9]徐明曜.有限群导引(上下册)[M].2版.北京:科学出版社,1999.