ON s-PERMUTABLY EMBEDDED AND WEAKLY c-NORMAL SUBGROUPS OF FINITE GROUPS

Year 2013, Volume: 14 Issue: 14 , 44 - 52 , 01.12.2013

Guo Zhong Liying Yang Xuanlong Ma Shixun Lin

https://izlik.org/JA73HX65PA

Abstract

Let G be a finite group, p the smallest prime dividing the order
of G and P a Sylow p-subgroup of G with the smallest generator number d.
We consider such a set Md(P) = {P1, P2, . . . , Pd} of maximal subgroups of P
such that ∩di=1Pi = Φ(P). Groups with certain s-permutably embedded and
weakly c-normal subgroups of prime power order are studied. We present some
sufficient conditions for a group to be p-nilpotent or p-supersolvable.

Keywords

weakly c-normal subgroups , s-permutably embedded subgroups , p-nilpotent groups

References

• K. Al-Sharo, On some maximal S-quasinormal subgroups of finite groups, Beitr¨age Algebra Geom., 49 (2008), 227-232.
• M. Asaad and A. A. Heliel, On S-quasinormal embedded subgroups of finite groups, J. Pure Appl. Algebra, 165 (2001), 129-135.
• A. Ballester-Bolinches and M. C. Pedraza-Aguilera, Sufficient conditions for supersolubility of finite groups, J. Pure Appl. Algebra, 127 (1998), 113-118.
• W. E. Deskins, On quasinormal subgroups of finite groups, Math. Z., 82 (1963), 132.
• D. Gorenstein, Finite Group, 2nd Edition. Chelsea Publishing Co., New York, X. Y. Guo and K. P. Shum, On c-normal maximal and minimal subgroups of Sylow p-subgroups of finite groups, Arch. Math., 80 (2003), 561-569.
• B. Huppert, Endliche Gruppen I, Die Grundlehren der Mathematischen Wis- senschaften, Band 134, Springer-Verlag, Berlin, 1967.
• O. H. Kegel, Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z., (1962), 205-221.
• S. Li and X. He, On normally embedded subgroups of prime power order in finite groups, Comm. Algebra, 36 (2008), 2333-2340.
• D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics vol. 80, Springer-Verlag, New York, 1982.
• P. Schmid, Subgroups permutable with all Sylow subgroups, J. Algebra, 207 (1998), 285-293.
• S.Srinivasan, Two sufficient conditions for supersolvability of finite groups, Is- rael J. Math., 35 (1980), 210-214.
• A. Skiba, On weakly s-permutable subgroups of finite groups, J. Algebra, 315 (2007), 192-209.
• J. G. Thompson, Normal p-complements for finite groups, J. Algebra, 1 (1964), 46.
• Y. Wang, c-normality of groups and its properties, J. Algebra, 180 (1996), 965. Y. Wang,
• Finite groups with some subgroups of Sylow subgroups c- supplemented, J. Algebra, 224 (2000), 464-478.
• H. Wei and Y. Wang, On c*-normality and its properties, J. Group Theory, 10 (2007), 211-223.
• L. Zhu, W. Guo and K. Shum, Weakly c-normal subgroup of finite groups and their properties, Comm. Algebra, 30 (2002), 5505-5512.
• Guo Zhong and Liying Yang School of Mathematical Sciences, Guangxi Teachers Education University, Nanning, P. R. China e-mails: [e-mail] (Guo Zhong) Xuanlong Ma
• School of Mathematical Sciences Beijing Normal University, Beijing, P. R. China e-mail: [e-mail] [e-mail] (Liying Yang) Shixun Lin
• College of Mathematics and Statistics, Zhaotong University, Zhaotong, P. R. China e-mail: [e-mail]