【摘要】：共軛類與有限群的結(jié)構(gòu)在群論的研究中有著重要的地位,在過去的幾十年中比較活躍,并取得了很多成果。本文主要利用共軛類長圖來研究有限群的結(jié)構(gòu)。共軛類長圖T(G)是滿足下面兩個(gè)條件的無向圖：(1)以群的非中心共軛類長的集合Cl(G)中的元素為頂點(diǎn)；(2)如果兩個(gè)頂點(diǎn)|Gi|和|Cj|之間有一條邊相連,當(dāng)且僅當(dāng)(|Ci|,|Cj|)1。利用共軛類長圖的頂點(diǎn)和邊的個(gè)數(shù)可以對(duì)有限群進(jìn)行分類。 在本文首先給出了頂點(diǎn)個(gè)數(shù)最多為4的圖,共有18個(gè)圖。根據(jù)群的共軛類長圖的定義以及共軛類長具有的一些性質(zhì),得到18個(gè)圖中只有8個(gè)圖可以作為有限群的共軛類長圖。對(duì)于這8個(gè)圖,得到了一些結(jié)果。若圖為一孤立點(diǎn),則群只有一個(gè)非中心共軛類長,此時(shí)群為其Sylow p子群與交換群的直積；若圖為兩個(gè)孤立點(diǎn),則群為擬Frobenius群。 通過GAP程序計(jì)算出群階在100以內(nèi)的群的共軛類長以及對(duì)應(yīng)群的結(jié)構(gòu),利用共軛類長圖給出了100階以內(nèi)的有限群的一個(gè)分類。 共軛類長圖是以一個(gè)群所有的共軛類長為研究對(duì)象來進(jìn)行考察的,而在研究過程中發(fā)現(xiàn)當(dāng)共軛類長滿足一定算數(shù)條件時(shí)也能得到一些有意義的結(jié)果。文中第四章定義了共軛類長的平方整除群階的群,即SCLD群,得到結(jié)論：(1)有限交換群為SCLD群；(2)單群、幾乎單群及Frobenius群不是SCLD群；(3)冪零群為SCLD群當(dāng)且僅當(dāng)它的Sylow p子群均為SCLD群。
[Abstract]:The structure of conjugate classes and finite groups plays an important role in the study of group theory and has been active in the past few decades and many achievements have been made. In this paper, the structure of finite groups is studied by using conjugate class graph. The conjugate class length graph T (G) is an undirected graph satisfying the following two conditions: (1) taking the elements in the set Cl (G) of the group's noncentral conjugate class length as vertices; (2) if there is an edge connected between the two vertices, if and only if (ci, Cj) 1. Finite groups can be classified by the number of vertices and edges of conjugate class graphs. In this paper, we first give a graph with the number of vertices up to 4, a total of 18 graphs. According to the definition of conjugate class length graph of group and some properties of conjugate class length, only 8 of 18 graphs can be regarded as conjugate class length graph of finite group. For these eight graphs, some results are obtained. If the graph is an isolated point, then the group has only one noncentral conjugate class length, where the group is the direct product of its Sylow p subgroup and abelian group, and if the graph is two isolated points, then the group is quasi Frobenius group. The conjugate class length and the structure of corresponding group are calculated by GAP program, and a classification of finite groups of order 100 is given by using conjugate class length graph. The conjugate class length graph is investigated by taking the conjugate class length of a group as the object of study, and it is found that some meaningful results can be obtained when the conjugate class length satisfies certain arithmetic conditions. In the fourth chapter, we define the group of square division group of conjugate class length, that is, SCLD group. The conclusions are as follows: (1) finite abelian group is SCLD group, (2) simple group, almost simple group and Frobenius group are not SCLD group; (3) Nilpotent groups are SCLD groups if and only if their Sylow p subgroups are SCLD groups.
【學(xué)位授予單位】：沈陽工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O152.1;O157.5

【參考文獻(xiàn)】

相關(guān)期刊論文 前10條

1 韋華全;田夢(mèng)飛;楊立英;張英杰;馬儇龍;;定義在有限群上的一類新的共軛類圖[J];廣西師范學(xué)院學(xué)報(bào)(自然科學(xué)版);2013年04期

2 游興中;;有限群的一類新的共軛類圖[J];長沙理工大學(xué)學(xué)報(bào)(自然科學(xué)版);2007年03期

3 畢建行;;有限單群共軛類的一個(gè)數(shù)量性質(zhì)[J];遼寧大學(xué)學(xué)報(bào)(自然科學(xué)版);2008年01期

4 朝魯,楊開宇,傅守忠;計(jì)算機(jī)代數(shù)學(xué)及其軟件系統(tǒng)[J];內(nèi)蒙古大學(xué)學(xué)報(bào)(自然科學(xué)版);1999年03期

5 羅馳,任永才;共軛類圖最多有三條邊的有限群[J];四川大學(xué)學(xué)報(bào)(自然科學(xué)版);2003年04期

6 邵長國;蔣琴會(huì);;具有給定共軛類長的有限群[J];上海大學(xué)學(xué)報(bào)(自然科學(xué)版);2011年05期

7 任永才;共軛類的長和有限群的結(jié)構(gòu)[J];數(shù)學(xué)進(jìn)展;1994年05期

8 游興中;錢國華;;一類和有限群的共軛類關(guān)聯(lián)的新圖[J];數(shù)學(xué)年刊A輯(中文版);2007年05期

9 陳生安;錢國華;;關(guān)于有限p-群特征標(biāo)值的一個(gè)注記(英文)[J];數(shù)學(xué)雜志;2012年04期

10 董井成;;GAP在近世代數(shù)教學(xué)中的應(yīng)用[J];高等數(shù)學(xué)研究;2011年01期

相關(guān)博士學(xué)位論文 前1條

1 孔慶軍;有限群的共軛類長及廣義正規(guī)子群[D];上海大學(xué);2009年

，

本文編號(hào)：2190520