最高阶元的个数与有限群结构

发布时间：2018-07-13 08:07

【摘要】：1963年Feit和J.G.Thompson教授证明了在群论方面闻名遐迩的奇阶定理,该定理指出:对于有限群而言,每一个奇数阶群都可解。而对于阶为偶数的群,J.G.Thompson教授在至施武杰教授的信函中,提出了经典的Thompson猜想。若Thompson猜想成立,我们则可以通过研究同阶型群的可解性去判断有限群的可解性。因此证明Thompson猜想的成立,对于研究有限群的可解性非常重要。1987年Thompson问题与猜想提出后,吸引了许多国内外群论工作者,他们主要在群的可解性与群的具体结构两方面做了许多研究,并取得了丰硕的成果。这为以后Thompson问题的研究提供了一定的指导思想和方法,为Thompson问题的解决奠定了一定的基础。对于Thompson猜想,虽没有学者找出一般性的判断方法,然而通过研究群可解性与有限群中最高阶元个数的特殊关系,可以从侧面对Thompson猜想在一些特殊条件下的具体情况进行一些研究,这也得出了许多可喜的结果,而对于Thompson猜想的证明这些结果至关重要,因此我将在这些结果的基础上,继续讨论最高阶元素个数为某些特殊数的有限群的可解性及具体的群结构。本文的主要结果:1.设G是有限群且最高阶元素为22个,则G可解,并为如下之一:(1)当k=6时,??G??32其中??????31,52。(2)当k={23,46}时,设x为k阶元素,则有Gxx CG?)(???,所以)()(k GGCAutx C?。2.设G是有限群且最高阶元素为28个,则G可解,并为如下之一:(1)当k=4时,2148??,,GGCQG或3G。(2)当k=6时,??G??32其中??????41,62。(3)当k=10时,GCS?55?,3255)(CCSCG??且4)(/5SCGG。(4)当k={29,58}时,设x为k阶元素,则有Gxx CG?)(???,所以)()(k GGCAutx C?。3.设G是有限群且最高阶元素为44个,则G可解,并为如下之一:(1)当k=4时,987654321?,,,,,,,,GGGGGGGGGG。(2)当k=6时,??G??32其中??????41,72。(3)当k={69,92,138}时,设x为k阶元素,则有Gxx CG?)(???,所以)()(k GGCAutx C?。综上,我们可以得出:当有限群的最高阶元素的个数为22、28、44时,Thompson猜想成立。
[Abstract]:In 1963, Professor Feit and J. G. Thompson proved the well-known odd order theorem in group theory. It is pointed out that every odd order group is solvable for a finite group. For the group with even order, Professor J. G. Thompson put forward the classical Thompson conjecture in his letter to Professor Schwujie. If Thompson's conjecture is true, we can judge the solvability of finite groups by studying the solvability of groups of the same order. Therefore, it is very important to prove that Thompson's conjecture is valid for studying the solvability of finite groups. After Thompson's problem and conjecture were put forward in 1987, it attracted many group theorists at home and abroad. They have mainly done a lot of research on the solvability of groups and the concrete structure of groups, and have achieved fruitful results. This provides a certain guiding ideology and method for the study of Thompson problem in the future, and lays a certain foundation for the solution of Thompson problem. For Thompson's conjecture, although no scholar has found a general judgment method, the special relationship between the solvability of groups and the number of the highest order elements in finite groups is studied. We can do some research on Thompson's conjecture under some special conditions from the side, which also leads to a lot of gratifying results, and it is very important for Thompson's conjecture to prove these results, so I will build on these results. We continue to discuss the solvability and the concrete group structure of finite groups with the highest order elements being some special numbers. The main result of this paper is: 1. Let G be a finite group with 22 elements of the highest order, then G is solvable and is one of the following: (1) if k = 6, where K = 6, where K = 31N, 522. (2) let x be an element of order k, then Gxx CG?) (so) (k GGCAutx CU. 2. Let G be a finite group with the highest order of 28 elements, then G is solvable and one of the following: (1) when k = 4, it is GGCQG or 3G. (2) when k = 6, it is GCQG or 3G. And 4) (/ 5 SCG GG. (4) when k = {2958}, let x be k order element, then there is Gxx CG?) (n) (therefore,) (k GGCAutx CU. 3. Let G be a finite group with 44 elements of the highest order, then G is solvable and is one of the following: (1) when k = 4, we have GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG@@ In summary, we can conclude that Thompson's conjecture is true when the number of the highest order elements of a finite group is 22 ~ 28 ~ 44 ~ 44.
【学位授予单位】：成都信息工程学院
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O152.1

【相似文献】