# zbMATH — the first resource for mathematics

$$S$$-semiembedded subgroups of finite groups. (English) Zbl 1328.20040
Summary: A subgroup $$H$$ of a finite group $$G$$ is said to be $$s$$-semipermutable in $$G$$ if it is permutable with every Sylow $$p$$-subgroup of $$G$$ with $$(p,|H|)=1$$. We say that a subgroup $$H$$ of a finite group $$G$$ is $$S$$-semiembedded in $$G$$ if there exists an $$s$$-permutable subgroup $$T$$ of $$G$$ such that $$TH$$ is $$s$$-permutable in $$G$$ and $$T\cap H\leqslant H_{\overline sG}$$, where $$H_{\overline sG}$$ is an $$s$$-semipermutable subgroup of $$G$$ contained in $$H$$. In this paper, we investigate the influence of $$S$$-semiembedded subgroups on the structure of finite groups.

##### MSC:
 20D40 Products of subgroups of abstract finite groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D15 Finite nilpotent groups, $$p$$-groups
Full Text:
##### References:
 [1] Chen, Z, On a theorem of srinivasan, J Southwest Normal Univ (Nat Sci), 12, 1-4, (1987) · Zbl 0732.20008 [2] Deskins, W E, On quasinormal subgroups of a finite group, Math Z, 82, 125-132, (1963) · Zbl 0114.02004 [3] Doerk K, Hawkes T. Finite Solvable Groups. Berlin: Walter de Gruyter, 1992 · Zbl 0753.20001 [4] Gorenstein D. Finite Groups. New York: Chelsea Publishing Co, 1968 · Zbl 0185.05701 [5] Guo W. The Theory of Classes of Groups. Beijing-New York: Science Press-Kluwer Academic Publishers, 2000 · Zbl 1005.20016 [6] Guo, W; Lu, Y; Niu, W, $$S$$-embedded subgroups of finite groups, Algebra Logic, 49, 293-304, (2010) · Zbl 1255.20021 [7] Guo, X; Shum, K P, On $$c$$-normal maximal and minimal subgroups of Sylow $$p$$-subgroup of finite groups, Arch Math, 80, 561-569, (2003) · Zbl 1050.20010 [8] Guo, W; Shum, K P; Skiba, A N, On solubility and supersolubility of some classes of finite groups, Sci China Ser A, 52, 272-286, (2009) · Zbl 1189.20023 [9] Huppert B. Endliche Gruppen I. New York: Springer, 1967 · Zbl 0217.07201 [10] Kegel, O H, Sylow-gruppen and subnormalteiler endlicher gruppen, Math Z, 78, 205-211, (1962) · Zbl 0102.26802 [11] Li, Y; Qiao, S; Su, N; Wang, Y, On weakly $$s$$-semipermutable subgroups of finite group, J Algebra, 371, 250-261, (2012) · Zbl 1269.20020 [12] Robinson D J S. A Course in Theory of Group. New York: Springer-Verlag, 1982 [13] Schmid, P, Subgroups permutable with all Sylow subgroups, J Algebra, 207, 285-293, (1998) · Zbl 0910.20015 [14] Skiba, A N, On weakly $$s$$-permutable subgroups of finite groups, J Algebra, 315, 192-209, (2007) · Zbl 1130.20019 [15] Srinivasan, S, Two sufficient conditions for supersolubility of finite groups, Israel J Math, 35, 210-214, (1980) · Zbl 0437.20012 [16] Wang, L; Wang, Y, On $$s$$-semipermutable maximal subgroups and minimal subgroups of Sylow $$p$$-subgroups of finite groups, Comm Algebra, 34, 143-149, (2006) · Zbl 1087.20015 [17] Wang, Y, $$C$$-normality of groups and its properties, J Algebra, 180, 954-961, (1996) · Zbl 0847.20010 [18] Wielandt H. Subnormal Subgroups and Permutation Groups. Lectures given at the Ohio State University, Columbia, Ohio, 1971 [19] Zhang, Q; Wang, L, The influence of $$s$$-semipermutable properties of subgroups on the structure of finite groups, Acta Math Sinica (Chin Ser), 48, 81-88, (2005) · Zbl 1119.20026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.