Finite Groups with some $s$-Permutably Embedded and Weakly $s$-Permutable Subgroups

Xie, Fenfang; Wang, Jinjin; Xia, Jiayi; Zhong, Guo

doi:10.5802/cml.4

Finite Groups with some

s

-Permutably Embedded and Weakly

s

-Permutable Subgroups

Xie, Fenfang ¹ ; Wang, Jinjin ¹ ; Xia, Jiayi ¹ ; Zhong, Guo ¹

¹ School of Mathematical Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China

Confluentes Mathematici, Tome 5 (2013) no. 1, pp. 93-101.

Résumé

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 $ℳ_{d} (P) = {P_{1}, P_{2}, \dots, P_{d}}$ 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.

Reçu le : 2012-05-10
Accepté le : 2013-04-16
Accepté après révision le : 2013-10-04
Publié le : 2017-03-27

DOI : 10.5802/cml.4

Classification : 20D10, 20D20
Mots-clés : weakly $s$-permutable subgoups; $s$-permutably embedded subgroups; $p$-nilpotent groups

Affiliations des auteurs :

Xie, Fenfang ¹ ; Wang, Jinjin ¹ ; Xia, Jiayi ¹ ; Zhong, Guo ¹

¹ School of Mathematical Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China

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     title = {Finite {Groups} with some $s${-Permutably} {Embedded} and {Weakly} $s${-Permutable} {Subgroups}},
     journal = {Confluentes Mathematici},
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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) no. 1, pp. 93-101. doi : 10.5802/cml.4. http://www.numdam.org/articles/10.5802/cml.4/

Bibliographie
Cité par

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

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

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

[4] K. Doerk, Finite Soluble Groups, Berlin, Walterde Gruyter, 1992.

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

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

[7] B. Huppert, Endliche gruppen I, Springer, Berlin, 1967.

[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.

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

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

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

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

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

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

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

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

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

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

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

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

[21] J. G. Thompson, Normal $p$ -complements for finite groups, J. Algebra, 1:43–46, 1964.

[22] Y. Wang, $c$ -normality of groups and its properties, J. Algebra, 180:954–965, 1996.

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

[24] H. Wei, Y. Wang, On $c^{*}$ -normality and its properties, J. Group Theory, 10:211–223, 2007.

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

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