本文選題：有限群 + 最高階元　； 參考：《西南大學(xué)》2017年碩士論文

【摘要】：在有限群的結(jié)構(gòu)研究中,用群的階,子群的階,元素的階得出了若干漂亮的群論性質(zhì).如:奠定有限群理論基礎(chǔ)的Sylow定理,描述子群階與群階性質(zhì)Lagrange定理,描述p | |G|時(shí),G有p階元存在的柯西定理,奇數(shù)階可解定理,等等.自上世紀(jì)80年代起,以施武杰教授為代表的群論專家開始關(guān)注用群的基本數(shù)量群的階和元素階之集來刻畫有限單群,并提出了著名的兩個(gè)階刻畫全部有限單群的猜想.在該猜想得到解決后,一些學(xué)者開始關(guān)注減少一些數(shù)量作為條件是否仍然可以刻畫單群.如:只用群的階和最高階元的階來刻畫單群,并得到很多單群的刻畫.本文試圖用Sylow 2-子群的階和最高階元的階來研究單群,但遺憾的是這種描述已不能成為刻畫.如:20階的Frobenius群的最高階元的階是5,2-Sylow子群的階是4,但恰A5也具有這兩個(gè)性質(zhì).基于此,本文從K3-單群的Sylow 2-子群和最高階元的階出發(fā),得出一些群論性質(zhì)的描述,由于水平有限,只給出了 8個(gè)K3-單群中7個(gè)單群對(duì)應(yīng)情況的性質(zhì),而且也無法得到”充要條件”描述,但作為本文的結(jié)論的推論,可以得”群的階”及”最高階元的階”能夠刻畫這7個(gè)單群.
[Abstract]:In the study of the structure of finite groups, some beautiful group theory properties are obtained by using the order of group, the order of subgroup and the order of element. For example, the Sylow theorem which lays the foundation of finite group theory, the Lagrange theorem of subgroup order and group order, the Cauchy theorem where p G has p order element, the odd order solvable theorem, and so on. Since the 1980s, the experts in group theory, represented by Professor Schwujie, have begun to focus on the description of finite simple groups by the order and element order of the basic number groups of groups, and have proposed a famous conjecture of two orders characterizing all finite simple groups. After the conjecture has been solved, some scholars begin to pay attention to whether reducing some quantity as a condition can still characterize a simple group. For example, only the order of a group and the order of the highest order are used to characterize a simple group, and many characterizations of a simple group are obtained. This paper attempts to study a simple group by using the order of Sylow 2-subgroup and the order of the highest order, but unfortunately this description can not be characterized. For example, the order of the highest order of a Frobenius group of order 20 is 4, but the order of a subgroup of 5 ~ 2-Sylow is 4, but A _ 5 also has these two properties. Based on this, from the order of Sylow 2-subgroup and the highest order of K3-simple group, some properties of group theory are obtained. Because of the limited level, we only give the corresponding properties of seven simple groups in eight K3-simple groups. But as a corollary of the conclusion of this paper, we can conclude that the order of a group and the order of the highest order can characterize these seven simple groups.
【學(xué)位授予單位】：西南大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O152.1

【參考文獻(xiàn)】

相關(guān)期刊論文 前10條

1 祿鵬;曹洪平;;元的階之集與群的性質(zhì)[J];西南大學(xué)學(xué)報(bào)(自然科學(xué)版);2013年12期

2 何立官;陳貴云;;關(guān)于一些對(duì)稱群的新刻畫[J];西南師范大學(xué)學(xué)報(bào)(自然科學(xué)版);2013年06期

3 何立官;陳貴云;;關(guān)于一些交錯(cuò)單群的新刻畫[J];重慶師范大學(xué)學(xué)報(bào)(自然科學(xué)版);2013年02期

4 何立官;陳貴云;;最高階元素個(gè)數(shù)為4pq的有限群[J];山西大學(xué)學(xué)報(bào)(自然科學(xué)版);2008年04期

5 晏燕雄;陳貴云;何立官;;最高階元素個(gè)數(shù)為52p的有限群[J];重慶大學(xué)學(xué)報(bào)(自然科學(xué)版);2006年09期

6 晏燕雄,陳貴云,何立官;最高階元個(gè)數(shù)為42的有限群是可解群[J];重慶師范大學(xué)學(xué)報(bào)(自然科學(xué)版);2005年03期

7 韓章家,陳貴云;最高階元素個(gè)數(shù)為2pq的有限群可解[J];西南師范大學(xué)學(xué)報(bào)(自然科學(xué)版);2004年02期

8 陳貴云;Frobenius群與2－Frobenius群的結(jié)構(gòu)[J];西南師范大學(xué)學(xué)報(bào)(自然科學(xué)版);1995年05期

9 施武杰,楊成;一類特殊的有限群[J];西南師范大學(xué)學(xué)報(bào)(自然科學(xué)版);1992年01期

10 施武杰;A_5的一個(gè)特征性質(zhì)[J];西南師范大學(xué)學(xué)報(bào)(自然科學(xué)版);1986年03期

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