本文關(guān)鍵詞： 交錯群 自同態(tài) 狀態(tài)空間圖 最小置換表示次數(shù)　出處：《西南大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：有限群為群論中非常重要的內(nèi)容,其結(jié)構(gòu)與性質(zhì)有著廣泛的應(yīng)用于.但由于這類研究的高度抽象性,在解決問題時往往需要從某些特殊的小階群入手.小階群由于結(jié)構(gòu)相對簡單,易于用相對淺顯和直觀的性質(zhì)來刻畫.本文用圖和次數(shù)分別對幾類小階群進行了研究.首先,利用狀態(tài)空間圖刻畫單群A5:設(shè)G為有限群,圖ΓG,α的頂點集為群G;對點集中的元素x,y,可以確定一條由x指向y的一條邊當(dāng)且僅當(dāng)α(x)=y.此時群G關(guān)于自同態(tài)α的狀態(tài)空間圖,記作ΓG,α.利用狀態(tài)空間圖進行討論,本文得到的主要結(jié)論如下:命題3.1 A5不能由它的二階自同構(gòu)誘導(dǎo)的狀態(tài)空間圖唯一刻畫.命題3.2 A5不能由它的四階自同構(gòu)誘導(dǎo)的狀態(tài)空間圖唯一刻畫.定理3.3設(shè)G是有限群,f為它的自同態(tài),δ是A5的三階自同構(gòu),若ΓG,f≌ΓA5,δ,則G≌A5.推論3.4設(shè)G是有限群,f為它的自同態(tài),θ是A5的六階自同構(gòu),若ΓG,f≌ΓA5,θ,則G≌A5.其次,主要研究了最小置換表示次數(shù).如果存在適當(dāng)?shù)恼麛?shù)d,使得G(?)Sd但G(?)Sd-1,則稱d為G的最小置換表示次數(shù),記作d(G).我們討論了56階及60階群到置換群的最小置換表示次數(shù).定理4.1根據(jù)56階群的分類[引理2.6]得到所有56階群的最小置換表示次數(shù)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根據(jù)60階群的分類[引理2.7]得到所有60階群的最小置換表示次數(shù)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 偽. Denote 螕 G, 偽. 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.
【學(xué)位授予單位】：西南大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O152

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