理论数学  >> Vol. 1 No. 2 (July 2011)

极大子群的h-完备与群的可解性
On h-Completion for Maximal Subgroups and the Solvability of Finite Groups

DOI: 10.12677/pm.2011.12025, PDF, HTML, 下载: 2,825  浏览: 11,897 

作者: 张晓燕, 徐伟华

关键词: 有限群θ-完备h-完备可解性
Finite Group; h-Completion; Solvability

摘要: 本文给出了有限群G极大子群M h-完备的定义,即称C为M的h-完备,若C是M的θ-完备,且有C = G或存在G的一个子群D使得D不是M的θ-完备,但有C < D成立. 从而,可以通过这个定义把θ-完备的极大性削弱,并得到了有关群可解性的重要结论,丰富了有限群理论。
Abstract: In this paper, we propose new definition of an h-completion for a maximal subgroup M of a group G, that is a θ-completion C such that either C = G or there exist a subgroup D of G which is not a θ-completion for M and C < D holds. This method can weaken the imposed maximality on θ-completion. Moreover, we studied the solvability and of finite groups by means of h-completion and obtained some important properties, which are very useful to research deeply finite groups.

文章引用: 张晓燕, 徐伟华. 极大子群的h-完备与群的可解性[J]. 理论数学, 2011, 1(2): 124-131. http://dx.doi.org/10.12677/pm.2011.12025

参考文献

[1] W. E. Deskin. On maximal subgroups. Proceedings of a Sympo-sium in Pure Mathematics of the American Mathematical Society, 1959, 1: 100-104.
[2] N. P. Mukherjee, P. Bhattachary. On theta pairs for maximal subgroups. Proceedings of the American Mathematical Society, 1990, 109(3): 589-596.
[3] Y. Q. Zhao. On the deskins completions, theta completions and theta pairs for maximal subgroups. Communications in Algebra, 1998, 26(2): 3141-3153.
[4] S. R. Li, Y. Q. Zhao. On s-completions of maximal subgroups of finite groups. Algebra Colloquium, 2004, 11(3): 315-324.
[5] Y. Q. Zhao. On the deskins completions, theta completions for maximal subgroups. Communications in Algebra, 2000, 28(1): 375-385.
[6] Y. Q. Zhao. On theta completions & s-completions for maximal subgroups. JP Journal of Algebra, Number Theory and Applications, 2002, 2(2): 111-119.
[7] 徐明耀. 有限群导引[M]. 北京: 科学出版社, 1999.
[8] T. Hawkes. On the class of Sylow Tower groups. Mathematische Zeitschrift, 1968, 105(5): 393-398.
[9] B. Huppert, G. I. Endliche. Finite groups. Berlin: Springer-Verlag, 1968.
[10] J. S. Rose. On finite in solute groups with nipotent maximal subgroups. Journal of Algebra, 1977, 48(4): 182-196.
[11] Y. Q. Zhao. Completion, theta-completion and the salvability of finite groups. Journal of Applied Algebra and Discrete Structures, 2003, 1(2): 87-97.
[12] Y. M. Wang. C-normality of groups and properties. Journal of Algebra, 1996, 180(3): 945-965.