Suppose that $H$ is a subgroup of $G$, then $H$ is said to be $s$-permutable in $G$, if $H$ permutes with every Sylow subgroup of $G$. If $HP=PH$ hold for every Sylow subgroup $P$ of $G$ with $(|P|, |H|)=1$), then $H$ is called an $s$-semipermutable subgroup of $G$. In this paper, we say that $H$ is partially $S$-embedded in $G$ if $G$ has a normal subgroup $T$ such that $HT$ is $s$-permutable in $G$ and $H\cap T\leq H_{\overline{s}G}$, where $H_{\overline{s}G}$ is generated by all $s$-semipermutable subgroups of $G$ contained in $H$. We investigate the influence of some partially $S$-embedded minimal subgroups on the nilpotency and supersolubility of a finite group $G$. A series of known results in the literature are unified and generalized.
A. Ballester-Bolinches and M. C. Pedraza-Aguilera (1996). On minimal
subgroups of finite group. Acta Math. Hungar.. 73, 335-342 A. Ballester-Bolinches and Y. M. Wang (2000). Finite groups with some
$c$-normal minimal subgroups. J. Pure Appl. Algebra. 153, 121-127 Z. M. Chen (1987). On a theorem of Srinivasan. J. Southwest Normal
Univ (Nat Sci). 12 (1), 1-1 J. B. Derr, W. E. Deskins and N. P. Mukherjee (1985). The influence of
minimal $p$-subgroups on the structure of finite groups. Arch. Math. (Basel). 45, 1-4 W. E. Deskins (1963). On quasinormal subgroups of finite groups. Math. Z.. 82 (2), 125-132 K. Doerk and T. Hawkes (1992). Finite Soluble Groups. Walter de
Gruyter, Berlin, New York. W. B. Guo (2000). The Theory of Classes of Groups. Science Press-Kluwer Academic Publishers, Beijing, New York, Dordrecht, Boston, London. W. B. Guo, K. P. Shum and A. N. Skiba (2009). On solubility and
supersolubility of some classes of finite groups. Sci. China
Ser. A. 52 (2), 272-286 W. B. Guo, K. P. Shum and F. Y. Xie (2011). Finite groups with some weakly
$s$-supplemented subgroups. Glasg. Math. J.. 53, 211-222 W. B. Guo, Y. Lu and W. Niu (2010). $S$-embedded subgroups of finite
groups. Algebra Logic. 49 (4), 293-304 B. Huppert (1967). Endliche Gruppen. I.. Springer-Verlag, Berlin, New York. B. Huppert and N. Blackburn (1982). Finite Groups III. Springer-Verlag, Berlin, New York. O. H. Kegel (1962). Sylow-Gruppen und Sbnormalteiler endlicher Gruppen. Math. Z.. 78, 205-221 Y. M. Li and Y. M. Wang (2004). On $\pi$-quasinormally embedded subgroups
of finite group. J. Algebra. 281, 109-123 D. J. S. Robinson (1993). A Course in the Theory of Groups. Springer-Verlag, Berlin, New York. A. N. Skiba (2007). On weakly $s$-permutable subgroups of finite groups. J. Algebra. 315, 192-209 Y. M. Wang (1996). $C$-normality of groups and its properties. J.
Algebra. 180, 954-965 Q. H. Zhang and L. F. Wang (2005). The influence of $s$-semipermutable
properties of subgroups on the structure of finite groups. Acta Math. Sinica Acta
Math. Sin.. 48 (1), 81-88
Zhao, T. and Zhang, Q. (2013). Partially $S$-embedded minimal subgroups of finite groups. International Journal of Group Theory, 2(4), 7-16. doi: 10.22108/ijgt.2013.2751
MLA
Zhao, T. , and Zhang, Q. . "Partially $S$-embedded minimal subgroups of finite groups", International Journal of Group Theory, 2, 4, 2013, 7-16. doi: 10.22108/ijgt.2013.2751
HARVARD
Zhao, T., Zhang, Q. (2013). 'Partially $S$-embedded minimal subgroups of finite groups', International Journal of Group Theory, 2(4), pp. 7-16. doi: 10.22108/ijgt.2013.2751
CHICAGO
T. Zhao and Q. Zhang, "Partially $S$-embedded minimal subgroups of finite groups," International Journal of Group Theory, 2 4 (2013): 7-16, doi: 10.22108/ijgt.2013.2751
VANCOUVER
Zhao, T., Zhang, Q. Partially $S$-embedded minimal subgroups of finite groups. International Journal of Group Theory, 2013; 2(4): 7-16. doi: 10.22108/ijgt.2013.2751
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