# zbMATH — the first resource for mathematics

Finite groups with given $$s$$-embedded and $$n$$-embedded subgroups. (English) Zbl 1182.20026
From the summary: “Let $$G$$ be a finite group and $$H$$ a subgroup of $$G$$. Then $$H$$ is said to be $$s$$-permutable if $$H$$ permutes with all Sylow subgroups of $$G$$. Let $$H_{sG}$$ be the subgroup generated by all subgroups of $$H$$ which are $$s$$-permutable in $$G$$ and let $$H^{sG}$$ be the intersection of all $$s$$-permutable subgroups of $$G$$ which contain $$H$$. We say that: (1) $$H$$ is $$s$$-embedded in $$G$$ if $$G$$ has an $$s$$-permutable subgroup $$T$$ such that $$T\cap H\leq H_{sG}$$ and $$HT=H^{sG}$$; (2) $$H$$ is $$n$$-embedded in $$G$$ if $$G$$ has a normal subgroup $$T$$ such that $$T\cap H\leq H_{sG}$$ and $$HT=H^G$$.
Our main results here are the following: Theorem A. A group $$G$$ is supersoluble if and only if every maximal subgroup of every non-cyclic Sylow subgroup of the generalized Fitting subgroup $$F^*(G)$$ of $$G$$ is $$n$$-embedded.
Theorem B. A group $$G$$ is supersoluble if and only if for every non-cyclic Sylow subgroup $$P$$ of the generalized Fitting subgroup $$F^*(G)$$ of $$G$$, every cyclic subgroup $$H$$ of $$P$$ with prime order and with order 4 (if $$P$$ is a non-Abelian 2-group and $$H\not\subseteq Z_\infty(G)$$) is $$n$$-embedded.
Theorem F. A group $$G$$ is supersoluble if and only if every $$2$$-maximal subgroup $$E$$ of $$G$$ with non-primary index $$|G:E|$$, both has a cyclic supplement in $$E^{sG}$$ and is $$s$$-embedded in $$G$$.”
A number of published results appear then as corollaries of these theorems.

##### 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 20D25 Special subgroups (Frattini, Fitting, etc.)
Full Text:
##### References:
 [1] Ore, O., Contributions in the theory of groups of finite order, Duke math. J., 5, 431-460, (1939) · JFM 65.0065.06 [2] Stonehewer, S.E., Permutable subgroups in infinite groups, Math. Z., 125, 1-16, (1972) · Zbl 0219.20021 [3] Doerk, K.; Hawkes, T., Finite soluble groups, (1992), Walter de Gruyter Berlin/New York · Zbl 0753.20001 [4] Skiba, Alexander N., On weakly s-permutable subgroups of finite groups, J. algebra, 315, 192-209, (2007) · Zbl 1130.20019 [5] Ito, N.; Szép, J., Uber die quasinormalteiler von endlichen gruppen, Act. sci. math., 23, 168-170, (1962) · Zbl 0112.02106 [6] Maier, R.; Schmid, P., The embedding of permutable subgroups in finite groups, Z. math., 131, 269-272, (1973) · Zbl 0259.20017 [7] Kegel, O., Sylow-gruppen and subnormalteiler endlicher gruppen, Math. Z., 78, 205-221, (1962) · Zbl 0102.26802 [8] Deskins, W.E., On quasinormal subgroups of finite groups, Math. Z., 82, 125-132, (1963) · Zbl 0114.02004 [9] Schmid, P., Subgroups permutable with all Sylow subgroups, J. algebra, 207, 285-293, (1998) · Zbl 0910.20015 [10] Gorenstein, D., Finite groups, (1968), Harper & Row Publishers New York/Evanston/London · Zbl 0185.05701 [11] Weinstein, M., Between nilpotent and solvable, (1982), Polygonal Publishing House Passaic, NJ · Zbl 0488.20001 [12] Huppert, B., Normalteiler and maximal untergruppen endlicher gruppen, Math. Z., 60, 409-434, (1954) · Zbl 0057.25303 [13] H. Wielandt, Subnormal subgroups and permutation groups, Lectures given at the Ohio State University, Columbus, Ohio, 1971 [14] Guo, W.; Shum, K.P.; Skiba, A.N., X-semipermutable subgroups of finite groups, J. algebra, 315, 31-41, (2007) · Zbl 1130.20017 [15] Agrawal, R.K., Generalized center and hypercenter of a finite group, Proc. amer. math. soc., 54, 13-21, (1976) · Zbl 0342.20011 [16] Ballester-Bolinches, A.; Ezquerro, L.M., Classes of finite groups, (2006), Springer Dordrecht · Zbl 0790.20032 [17] Huppert, B., Endliche gruppen I, (1967), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0217.07201 [18] Shemetkov, L.A., Formations of finite groups, (1978), Nauka, Main Editorial Board for Physical and Mathematical Literature Moscow · Zbl 0496.20014 [19] Huppert, B.; Blackburn, N., Finite groups III, (1982), Springer-Verlag Berlin/New York · Zbl 0514.20002 [20] Buckley, J., Finite groups whose minimal subgroups are normal, Math. Z., 15, 15-17, (1970) · Zbl 0202.02303 [21] Srinivasan, S., Two sufficient conditions for supersolvability of finite groups, Israel J. math., 35, 210-214, (1980) · Zbl 0437.20012 [22] Wang, Y., c-normality of groups and its properties, J. algebra, 180, 954-965, (1996) · Zbl 0847.20010 [23] Ballester-Bolinches, A.; Wang, Y., Finite groups with some C-normal minimal subgroups, J. pure appl. algebra, 153, 121-127, (2000) · Zbl 0967.20009 [24] 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 [25] Ballester-Bolinches, A.; Pedraza-Aguilera, M.C., On minimal subgroups of finite groups, Acta math. hungar., 73, 335-342, (1996) · Zbl 0930.20021 [26] 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 [27] 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 [28] Wei, H., On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups, Comm. algebra, 29, 2193-2200, (2001) · Zbl 0990.20012 [29] 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 [30] 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 [31] Asaad, M.; Csörgő, P., Influence of minimal subgroups on the structure of finite group, Arch. math. (basel), 72, 401-404, (1999) · Zbl 0938.20013 [32] Li, Y.; Wang, Y., The influence of minimal subgroups on the structure of a finite group, Proc. amer. math. soc., 131, 337-341, (2002) [33] 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 [34] Guo, Wenbin, The theory of classes of groups, (2000), Science Press/Kluwer Academic Publishers Beijing/New York/Dordrecht/Boston/London · Zbl 1005.20016 [35] Asaad, M., On the solvability of finite groups, Arch. math. (basel), 51, 289-293, (1988) · Zbl 0656.20031 [36] Asaad, M., On maximal subgroups of finite group, Comm. algebra, 26, 3647-3652, (1998) · Zbl 0915.20008
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.