小阶群的状态空间图和最小置换表示次数

发布时间：2018-02-14 17:12

本文关键词： 交错群自同态状态空间图最小置换表示次数　出处：《西南大学》2017年硕士论文　论文类型：学位论文

【摘要】：有限群为群论中非常重要的内容,其结构与性质有着广泛的应用于.但由于这类研究的高度抽象性,在解决问题时往往需要从某些特殊的小阶群入手.小阶群由于结构相对简单,易于用相对浅显和直观的性质来刻画.本文用图和次数分别对几类小阶群进行了研究.首先,利用状态空间图刻画单群A5:设G为有限群,图ΓG,α的顶点集为群G;对点集中的元素x,y,可以确定一条由x指向y的一条边当且仅当α(x)=y.此时群G关于自同态α的状态空间图,记作ΓG,α.利用状态空间图进行讨论,本文得到的主要结论如下:命题3.1 A5不能由它的二阶自同构诱导的状态空间图唯一刻画.命题3.2 A5不能由它的四阶自同构诱导的状态空间图唯一刻画.定理3.3设G是有限群,f为它的自同态,δ是A5的三阶自同构,若ΓG,f≌ΓA5,δ,则G≌A5.推论3.4设G是有限群,f为它的自同态,θ是A5的六阶自同构,若ΓG,f≌ΓA5,θ,则G≌A5.其次,主要研究了最小置换表示次数.如果存在适当的正整数d,使得G(?)Sd但G(?)Sd-1,则称d为G的最小置换表示次数,记作d(G).我们讨论了56阶及60阶群到置换群的最小置换表示次数.定理4.1根据56阶群的分类[引理2.6]得到所有56阶群的最小置换表示次数如下:d(G1)= 15,d(G2)= 13,d(G3)= 13,d(G4)= 15,d(G5)= 11,d(G6)= 15,d(G7)＝11,d(G8)= 15,d(G9)= 11,d(G10)= 7,d(G11)= 15,d(G12)= 13,d(G13)= 8.定理4.2根据60阶群的分类[引理2.7]得到所有60阶群的最小置换表示次数如下:d(H1)= 12,d(H2)= 10,d(H3)= 10,d(H4)= 10,d(H5)= 9,d(H6)= 8,d(H7)=5,d(H8)= 9,d(H9)= 10,d(H10)= 12,d(H11)= 9.
[Abstract]:Finite group is a very important content in group theory, and its structure and properties are widely used. However, due to the high abstraction of this kind of research, In order to solve the problem, we often need to start with some special small order groups. Because of their relatively simple structure, small order groups are easy to be characterized by relatively simple and intuitive properties. In this paper, some small order groups are studied by graphs and times, respectively. Let G be a finite group, let the vertex set of graph 螕 G be a group G, and for the element of a point set XY, we can determine an edge from x to y if and only if a group G is a state space graph of an endomorphism 伪. Denote 螕 G, 伪. Discuss by using state space graph, The main conclusions of this paper are as follows: proposition 3.1 A 5 cannot be uniquely characterized by its state space graph induced by its second order automorphism. Proposition 3.2 A 5 cannot be uniquely characterized by its state space graph induced by its fourth order automorphism. Theorem 3.3. G is the finite group f is its endomorphism, 未 is the third order automorphism of A5, Let G be a finite group f be its endomorphism, 胃 be the sixth order automorphism of A 5. If 螕 GF = 螕 A 5, 胃, then G = A 5. Secondly, we mainly study the minimum permutation representation degree. If there is an appropriate positive integer d, such that G is? SD, but Gon? Sd-1, then d is the minimum permutation representation of G, We discuss the minimum permutation representation times of groups of order 56 and order 60 to permutation groups. Theorem 4.1 according to the classification of groups of order 56 [Lemma 2.6], we obtain the minimum permutation representation of all groups of order 56 as follows: DU G1 = 15 DU G2U = 13 DU G4 = 15du G5 = 11 DU G6U = 15dG7U G8 = 15dG9 = 11dG10. Theorem 4.2 according to the classification of groups of order 60 [Lemma 2.7], the minimum permutation times of all groups of order 60 are obtained as follows: dHH1 = 12 dldH2 + + = 10dH4 = 10dH4 = 10dH7dH6 = 8dH7 = 9dH9 = 10ddH10 = 12dH11 = 9.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O152

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