Finite Groups with some s-Permutably Embedded and Weakly s-Permutable Subgroups
Xie, Fenfang ; Wang, Jinjin ; Xia, Jiayi ; Zhong, Guo
Confluentes Mathematici, Tome 5 (2013), p. 93-100 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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. There is a set ℳ d (P)={P 1 ,P 2 ,⋯,P d } of maximal subgroups of P such that ⋂ i=1 d P i =Φ(P). In the present paper, we investigate the structure of a finite group under the assumption that every member of ℳ d (P) is either s-permutably embedded or weakly s-permutable in G to give criteria for a group to be p-supersolvable or p-nilpotent.

Publié le : 2013-01-01
DOI : https://doi.org/10.5802/cml.4
Classification:  20D10,  20D20 
@article{CML_2013__5_1_93_0,
     author = {Xie, Fenfang and Wang, Jinjin and Xia, Jiayi and Zhong, Guo},
     title = {Finite Groups with some $s$-Permutably Embedded and Weakly $s$-Permutable Subgroups},
     journal = {Confluentes Mathematici},
     volume = {5},
     year = {2013},
     pages = {93-100},
     doi = {10.5802/cml.4},
     mrnumber = {3143613},
     language = {en},
     url = {http://dml.mathdoc.fr/item/CML_2013__5_1_93_0}
}

Xie, Fenfang; Wang, Jinjin; Xia, Jiayi; Zhong, Guo. Finite Groups with some $s$-Permutably Embedded and Weakly $s$-Permutable Subgroups. Confluentes Mathematici, Tome 5 (2013) pp. 93-100. doi : 10.5802/cml.4. http://gdmltest.u-ga.fr/item/CML_2013__5_1_93_0/

[1] K. Al-Sharo, On some maximal S-quasinormal subgroups of finite groups, Beiträge zur Algebra und Geometrie, 49:227–232, 2008. | MR 2410577 | Zbl 1144.20009

[2] M. Asaad, A. A. Heliel, On S-quasinormal embedded subgroups of finite groups, J. Pure Appl. Algebra, 165:129–135, 2001. | MR 1865961 | Zbl 1011.20019

[3] A. Ballester-Bolinches, M. C. Pedraza-Aguilera, Sufficient conditions for supersolubility of finite groups, J. Pure Appl. Algebra, 127:113–118, 1998. | MR 1620696 | Zbl 0928.20020

[4] K. Doerk, Finite Soluble Groups, Berlin, Walterde Gruyter, 1992. | MR 1169099 | Zbl 0753.20001

[5] W. E. Deskins, On quasinormal subgroups of finite groups, Mathematische Zeitschrift, 82:125–132, 1963. | MR 153738 | Zbl 0114.02004

[6] D. Gorenstein, Finite group, Chelsea, New York, 1980.

[7] B. Huppert, Endliche gruppen I, Springer, Berlin, 1967. | MR 224703 | Zbl 0412.20002

[8] X. He, S. Li, X. Liu, On S-quasinormal and c-normal subgroups of prime power order in finite groups, Algebra Colloq., 18 (2011), 685–692. | MR 2837005 | Zbl 1241.20027

[9] O. H. Kegel, Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z., 78:205–221, 1962. | MR 147527 | Zbl 0102.26802

[10] S. Li, X. He, On normally embedded subgroups of prime power order in finite groups, Comm. Algebra, 36:2333–2340, 2008. | MR 2418390 | Zbl 1146.20015

[11] S. Li, Y. Li, On S-quasinormal and c-normal subgroups of a finite group, Czechoslovak Mathematical Journal 58:1083–1095, 2008. | MR 2471167 | Zbl 1166.20013

[12] S. Li, Z. Shen, J. Liu, et al, The influence of SS-quasinormality of some subgroups on the structure of finite groups, J. Algebra, 319:4275–4287, 2008. | MR 2407900 | Zbl 1152.20019

[13] D. H. Mclain, The existence of subgroups of given order in finite groups, Proc.Cambridge Philos.Soc, 53:278–285, 1957. | MR 85260 | Zbl 0079.25403

[14] L. Miao, On weakly s-permutable subgroups of finite groups, Bull Braz. Math. Soc. New Series, 41:223–235, 2010. | MR 2738912 | Zbl 1221.20014

[15] D. J. S. Robinson, A course in the Theory of groups, New York, Springer-Verlag, 1982. | MR 648604 | Zbl 0836.20001

[16] Z. Shen, W. Shi, Q. Zhang, S-quasinormality of finite groups, Front. Math. China, 5:329–339, 2010. | MR 2610728 | Zbl 1200.20014

[17] P. Schmidt, Subgroups permutable with all Sylow subgroups, J. Algebra, 207:285–293, 1998. | MR 1643106 | Zbl 0910.20015

[18] X. Shen, S. Li, W. Shi, Finite groups with normally embedded subgroups, J. Group Theory, 13:257–265, 2010. | MR 2607580 | Zbl 1196.20022

[19] A. N. Skiba, On weakly s-permutable subgroups of finite groups, J. Algebra, 315:192–209, 2007. | MR 2344341 | Zbl 1130.20019

[20] S. Srinivasan, Two sufficient conditions for supersolvability of finite groups, Israel J.Math, 35:210–214, 1980. | MR 576471 | Zbl 0437.20012

[21] J. G. Thompson, Normal p-complements for finite groups, J. Algebra, 1:43–46, 1964. | MR 167521 | Zbl 0119.26802

[22] Y. Wang, c-normality of groups and its properties, J. Algebra, 180:954–965, 1996. | MR 1379219 | Zbl 0847.20010

[23] Y. Wang, Finite groups with some subgroups of Sylow subgroups c-supplemented, J. Algebra, 224:464-478, 2000. | MR 1739589 | Zbl 0953.20010

[24] H. Wei, Y. Wang, On c * -normality and its properties, J. Group Theory, 10:211–223, 2007. | MR 2302616 | Zbl 1125.20011

[25] H. Wielandt, Subnormal subgroups and permutation groups, lectures given at the Ohio State University, Columbus, Ohio, 1971.