# zbMATH — the first resource for mathematics

On weakly $$s$$-permutable subgroups of finite groups. (English) Zbl 1130.20019
A subgroup $$H$$ in a finite group $$G$$ is weakly $$s$$-permutable in $$G$$ if there exists a subnormal subgroup $$T$$ such that $$G=HT$$ and $$T\cap H\subseteq H_{sG}$$, $$H_{sG}$$ the subgroup of $$H$$ generated by all subgroups of $$H$$ that are $$s$$-permutable in $$G$$. Consequently those classification results which require a subgroup to be either $$c$$-normal or $$s$$-permutable may be nontrivially generalized on the basis of being weakly $$s$$-permutable.
The author focuses on the prominent attributes of being weakly $$s$$-permutable underlying the hypotheses of twenty-four published results which become corollaries to the two principal theorems, a remarkable achievement that highlights the insight given to the development of the concept. The key is to fix in each noncyclic Sylow subgroup $$P$$ of $$G$$ a subgroup $$D$$ that satisfies $$1<|D|<|P|$$ and to study the structure of $$G$$ under the assumption that each subgroup $$H$$ with $$|H|=|D|$$ is weakly $$s$$-permutable.

##### MSC:
 20D40 Products of subgroups of abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure
Full Text:
##### References:
 [1] Kegel, O., Sylow-gruppen and subnormalteiler endlicher gruppen, Math. Z., 78, 205-221, (1962) · Zbl 0102.26802 [2] Wang, Y., c-normality of groups and its properties, J. algebra, 180, 954-965, (1996) · Zbl 0847.20010 [3] Ballester-Bolinches, A.; Esteban-Romero, R., On finite groups in which Sylow permutability is a transitive relation, Acta math. hungar., 101, 193-202, (2003) · Zbl 1064.20022 [4] Buckley, J., Finite groups whose minimal subgroups are normal, Math. Z., 15, 15-17, (1970) · Zbl 0202.02303 [5] Laue, R., Dualization for saturation for locally defined formations, J. algebra, 52, 347-353, (1978) · Zbl 0383.20015 [6] Srinivasan, S., Two sufficient conditions for supersolvability of finite groups, Israel J. math., 35, 210-214, (1980) · Zbl 0437.20012 [7] Ramadan, M., Influence of normality on maximal subgroups of Sylow subgroups of a finite group, Acta math. hungar., 59, 107-110, (1992) · Zbl 0802.20019 [8] Li, D.; Guo, X., The influence of c-normality of subgroups on the structure of finite groups, II, Comm. algebra, 26, 1913-1922, (1998) · Zbl 0906.20012 [9] Wei, H., On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups, Comm. algebra, 29, 2193-2200, (2001) · Zbl 0990.20012 [10] Wei, H.; Wang, Y.; Li, Y., On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups, II, Comm. algebra, 31, 4807-4816, (2003) · Zbl 1050.20011 [11] Ballester-Bolinches, A.; Wang, Y., Finite groups with some C-normal minimal subgroups, J. pure appl. algebra, 153, 121-127, (2000) · Zbl 0967.20009 [12] Skiba, A.N., A note on c-normal subgroups of finite groups, Algebra discrete math., 3, 85-95, (2005) · Zbl 1092.20018 [13] J.J. Jaraden, A.N. Skiba, On c-normal subgroups of finite groups, Comm. Algebra, in press · Zbl 1137.20014 [14] Shaalan, A., The influence of π-quasinormality of some subgroups on the structure of a finite group, Acta math. hungar., 56, 287-293, (1990) · Zbl 0725.20018 [15] Asaad, M.; Ramadan, M.; Shaalan, A., Influence of π-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroups of a finite group, Arch. math. (basel), 56, 521-527, (1991) · Zbl 0738.20026 [16] Asaad, M., On the solvability of finite groups, Arch. math. (basel), 51, 289-293, (1988) · Zbl 0656.20031 [17] Asaad, M.; Csörgő, P., Influence of minimal subgroups on the structure of finite group, Arch. math. (basel), 72, 401-404, (1999) · Zbl 0938.20013 [18] Asaad, M., On maximal subgroups of finite group, Comm. algebra, 26, 3647-3652, (1998) · Zbl 0915.20008 [19] Li, Y.; Wang, Y., The influence of minimal subgroups on the structure of a finite group, Proc. amer. math. soc., 131, 337-341, (2002) [20] Li, Y.; Wang, Y., The influence of π-quasinormality of some subgroups of a finite group, Arch. math. (basel), 81, 245-252, (2003) · Zbl 1053.20017 [21] Ballester-Bolinches, A.; Pedraza-Aguilera, M.C., On minimal subgroups of finite groups, Acta math. hungar., 73, 335-342, (1996) · Zbl 0930.20021 [22] Guo, W.; Shum, K.P.; Skiba, A.N., G-covering subgroup systems for the classes of supersoluble and nilpotent groups, Israel J. math., 138, 125-138, (2003) · Zbl 1050.20009 [23] Alsheik Ahmad, A., Finite groups with given c-permutable subgroups, Algebra discrete math., 2, 9-16, (2004) · Zbl 1067.20018 [24] Miao, L.; Guo, W., Finite groups with some primary subgroups $$\mathcal{F}$$-s-supplemented, Comm. algebra, 33, 2789-2800, (2005) · Zbl 1084.20014 [25] Gorenstein, D., Finite groups, (1968), Harper & Row Publishers New York · Zbl 0185.05701 [26] Shemetkov, L.A., Formations of finite groups, (1978), Nauka, Main Editorial Board for Physical and Mathematical Literature Moscow · Zbl 0496.20014 [27] Huppert, B., Endliche gruppen I, (1967), Springer-Verlag Berlin · Zbl 0217.07201 [28] Doerk, K.; Hawkes, T., Finite soluble groups, (1992), Walter de Gruyter Berlin · Zbl 0753.20001 [29] H. Wielandt, Subnormal subgroups and permutation groups, Lectures given at the Ohio State University, Columbus, Ohio, 1971 [30] Schmid, P., Subgroups permutable with all Sylow subgroups, J. algebra, 82, 285-293, (1998) · Zbl 0910.20015 [31] Huppert, B.; Blackburn, N., Finite groups III, (1982), Springer-Verlag Berlin · Zbl 0514.20002 [32] Ballester-Bolinches, A.; Ezquerro, L.M., Classes of finite groups, (2006), Springer-Verlag Dordrecht · Zbl 0790.20032 [33] Sergienko, V.I., A criterion for the p-solubility of finite groups, Mat. zametki, Math. notes, 9, 216-220, (1971), (in Russian) · Zbl 0232.20023
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.