某些子群为SSH-子群的有限群
On Finite Groups of SSH-Subgroups

作者: 梁坚全 :广西大学,广西 南宁;

关键词: SSH-子群p-幂零群Sylow p-子群SSH-Subgroups p-Nilpotent Groups Sylow p-Subgroups

摘要:

设H是群G的一个子群,如果存在G的一个s-置换子群K,使得HsG=HK并且对任意g∈G都有Hg∩NK(H)≤H成立,则称H为G的SSH-子群。其中HsG是G的包含着H的最小的s-置换子群。文章研究了具有素数幂阶SSH-子群的有限群的结构,给出了有限群为p-幂零群的一些刻画条件。

Abstract: Let G be a fnite group. A subgroup H of G is s-permutable in G if H permutes with every Sylow subgroup of G. A subgroup H of G is called an SSH-subgroup in G if G has an s-permutable subgroup K such that HsG=HK and Hg∩NK(H)≤H, for all g∈G,where HsG is the intersection of all s-permutable subgroups of G containing H. This article studies the structure of fnite groups with SSH-subgroup which is prime power order. Some characterizations of a fnite group as a p-nilpotent group are given.

文章引用: 梁坚全 (2020) 某些子群为SSH-子群的有限群。 理论数学, 10, 30-37. doi: 10.12677/PM.2020.101006

参考文献

[1] Bianchi, M., Mauri, A.G.B., Herzog, M. and Verardi, L. (2000) On Finite Solvable Groups inWhich Normality Is a Transitive Relation. Journal of Group Theory, 3, 147-156.
https://doi.org/10.1515/jgth.2000.012

[2] Asaad, M., Heliel, A.A. and Al-Mosa Al-Shomrani, M.M. (2012) On Weakly H-Subgroups of Finite Groups. Communications in Algebra, 40, 3540-3550.
https://doi.org/10.1080/00927872.2011.591218

[3] Asaad, M. and Ramadan, M. (2016) On Weakly H-Embedded Subgroups of Finite Groups.Communications in Algebra, 44, 4564-4574.
https://doi.org/10.1080/00927872.2015.1130139

[4] Wei, X. and Guo, X. (2012) On HC-Subgroups and the Structure of Finite Groups. Commu-nications in Algebra, 40, 3245-3256.
https://doi.org/10.1080/00927872.2011.565846

[5] Asaad, M. and Ramadan, M. (2016) On Weakly HC-Embedded Subgroups of Finite Groups.Journal of Algebra and Its Applications, 15, Article ID: 1650077.
https://doi.org/10.1142/S0219498816500778

[6] Al-Gafri, T.M. and Nauman, S.K. (2018) On SSH-Subgroups of Finite Groups. Annali dell.Universita di Ferrara, 64, 209-225.
https://doi.org/10.1007/s11565-018-0299-1

[7] Ballester-Bolinches, A. and Esteban-Romero, R. (2003) On Finite T -Groups. Journal of the Australian Mathematical Society, 75, 181-191.
https://doi.org/10.1017/S1446788700003712

[8] Guo, X.Y. and Shum, K.P. (2003) Cover-Avoidance Properties and the Structure of Finite Groups. Journal of Pure and Applied Algebra, 181, 297-308.
https://doi.org/10.1016/S0022-4049(02)00327-4

[9] Kurzweil, H. and Stellmacher, B. (2004) The Theory of Finite Groups An Introduction.Springer-Universitext, New York-Berlin-Heidelberg-Hong Kong-London-Milan-Paris-Tokyo.
https://doi.org/10.1007/b97433

[10] Gorenstein, D. (1980) Finite Groups. Chelsea, New York.

[11] Guo, W., Shum, K.P. and Skiba, A.N. (2004) G-Covering Systems of Subgroups for Classes ofp-Supersoluble and p-Nilpotent Finite Groups. Siberian Mathematical Journal, 45, 433-442.
https://doi.org/10.1023/B:SIMJ.0000028608.59920.af

[12] Weinstein, M. (1982) Between Nilpotent and Solvable. Polygonal Publishing House, Passaic,NJ.

[13] Huppert, B. (1967) Endliche Gruppen I. Springer, Berlin.
https://doi.org/10.1007/978-3-642-64981-3

[14] Robinson, D.J.S. (1993) A Course in the Theory of Groups. Spring-Verlag, New York-Berlin.

分享
Top