On weakly $SS$-quasinormal and hypercyclically embedded properties of finite groups

Document Type : Research Paper


School of Science, Shandong University of Technology


A subgroup $H$ is said to be $s$-permutable in a group $G$‎, ‎if‎ ‎$HP=PH$ holds for every Sylow subgroup $P$ of $G$‎. ‎If there exists a‎ ‎subgroup $B$ of $G$ such that $HB=G$ and $H$ permutes with every‎ ‎Sylow subgroup of $B$‎, ‎then $H$ is said to be $SS$-quasinormal in‎ ‎$G$‎. ‎In this paper‎, ‎we say that $H$ is a weakly $SS$-quasinormal‎ ‎subgroup of $G$‎, ‎if there is a normal subgroup $T$ of $G$ such that‎ ‎$HT$ is $s$-permutable and $H\cap T$ is $SS$-quasinormal in $G$‎. ‎By‎ ‎assuming that some subgroups of $G$ with prime power order have the‎ ‎weakly $SS$-quasinormal properties‎, ‎we get some new‎ ‎characterizations about the hypercyclically embedded subgroups of‎ ‎$G$‎. ‎A series of known results in the literature are unified and‎ ‎generalized.


Main Subjects

A. Ballester-Bolinches and M. C. Pedraza-Aguilera (1998). Sufficient conditions for supersolvability of finite groups. J. Pure Appl. Algebra. 127, 113-118 A. Ballester-Bolinches, Y. Wang and X. Guo (2000). C-supplemented subgroups of finite groups. Glasg. Math. J.. 42, 383-389 W. E. Deskins (1963). On quasinormal subgroups of finite groups. Math. Z.. 82, 125-132 K. Doerk and T. Hawkes (1992). Finite Soluble Groups. Walter de Gruyter, Berlin, New York. D. Gorenstein (1968). Finite Groups. Chelsea, New York-London. W. Guo (2000). The Theory of Classes of Groups. Science Press-Kluwer Academic Publishers, New York. W. Guo, Y. Wang and L. Shi (2008). On Nearly s-normal subgroups of finite a group. J. Algebra Discrete Struct.. 6 (2), 95-106 W. 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. Guo, Y. Lu and W. Niu (2010). S-embedded subgroups of finite groups. Algebra Log.. 49 (4), 293-304 B. Huppert (1967). Endliche Gruppen Vol. I. Springer, New York, Berlin. O. H. Kegel (1962). Sylow-Gruppen und Subnormalteiler endlicher Gruppen. Math. Z.. 78, 205-221 S. Li, Z. Shen, J. Liu and X. Liu (2008). The influence of SS-quasinormality of some subgroups on the structure of finite groups. J. Algebra. 319, 4275-4287 Y. Li and X. Li (2005). Z-permutable subgroups and p-nilpotency of finite groups. J. Pure Appl. Algebra. 202, 72-81 Y. Li, S. Qiao and Y. Wang (2009). On weakly s-permutably embedded subgroups of finite groups. Commun. Algebra. 37, 1086-1097 A. N. Skiba (2007). On weakly s-permutable subgroups of finite groups. J. Algebra. 315, 192-209 A. N. Skiba (2011). A characterization of the hypercyclically embedded subgroups of finite group. J. Pure Appl. Algebra. 215, 257-261 P. Schmid (1998). Subgroups permutable with all Sylow subgroups. J. Algebra. 207, 285-293 Y. Wang (1996). C-normality of groups and its properties. J. Algebra. 180, 954-965 Y. Wang and W. Guo (2010). Nearly s-normality of groups and its properties. Comm. Algebra. 38, 3821-3836 H. Wei and Y. Wang (2007). The c-supplemented property of finite groups. Proc. Edinb. Math. Soc. (2). 50, 493-508 H. Wei and Y. Wang (2007). On c^{*}-normality and its properties. J. Group Theory. 10, 211-223 H. G. Bray and M. Weinstein (1982). Between nilpotent and solvable. Polygonal Publishing House.