Abstract
We introduce a new subgroup embedding property in a finite group called weakly S-quasinormality. We say a subgroup H of a finite group G is weakly S-quasinormal in G if there exists a normal subgroup K such that HK ⊴ G and H ∩ K is S-quasinormally embedded in G. We use the new concept to investigate the properties of some finite groups. Some previously known results are generalized.
[1] ASAAD, M.: On maximal subgroups of finite group, Comm. Algebra 26 (1998), 3647–3652. http://dx.doi.org/10.1080/0092787980882636410.1080/00927879808826364Search in Google Scholar
[2] ASAAD, M.— HELIEL, A. A.: On S-quasinormally embedded subgroups of finite groups, J. Pure Appl. Algebra 165 (2001), 129–135. http://dx.doi.org/10.1016/S0022-4049(00)00183-310.1016/S0022-4049(00)00183-3Search in Google Scholar
[3] ASAAD, M.— RAMADAN, M.— SHAALAN, A.: Influence of π-quasinormality on maximal subgroups of Sylow sub-groups of Fitting subgroups of a finite group, Arch. Math. (Basel) 56 (1991), 521–527. http://dx.doi.org/10.1007/BF0124676610.1007/BF01246766Search in Google Scholar
[4] BALLESTER, B. A.— PEDRAZA, A. M. C.: Sufficient conditions for supersolvability of finite groups, J. Pure Appl. Algebra 127 (1998), 113–118. http://dx.doi.org/10.1016/S0022-4049(96)00172-710.1016/S0022-4049(96)00172-7Search in Google Scholar
[5] BALLESTER, B. A.— PEDRAZA, A. M. C.: On minimal subgroups of finite groups, Acta Math. Hungar. 73 (1996), 335–342. http://dx.doi.org/10.1007/BF0005290910.1007/BF00052909Search in Google Scholar
[6] BALLESTER, B. A.— WANG, Y.: Finite groups with some c-normal minimal subgroups, J. Pure Appl. Algebra 153 (2000), 121–127. http://dx.doi.org/10.1016/S0022-4049(99)00165-610.1016/S0022-4049(99)00165-6Search in Google Scholar
[7] CHEN, S.— GUO, W.: S-c-permutably embedded subgroups of finite groups, Int. J. Contemp. Math. Sci. 3 (2008), 951–960. Search in Google Scholar
[8] DESKINS, W. E.: On quasinormal subgroups of finite groups, Math. Z. 82 (1963), 125–132. http://dx.doi.org/10.1007/BF0111180110.1007/BF01111801Search in Google Scholar
[9] DOREK, K.— HAWKES, T.: Finite Soluble Groups, Water de Gruyter, Berlin-New York, 1992. http://dx.doi.org/10.1515/978311087013810.1515/9783110870138Search in Google Scholar
[10] GORENSTEIN, D.: Finite Groups, Harper and Row Publishers, New York-Evanston-London, 1968. Search in Google Scholar
[11] GUO, W.: The Theory of Classes of Groups Math. Appl. 505, Kluwer Academic Publishers/Science Press, Dordrecht/Beijing, 2000. Search in Google Scholar
[12] HUPPERT, B.: Endlich Gruppen I, Springer-Verlag, Berlin-Heidelberg-New York, 1967. http://dx.doi.org/10.1007/978-3-642-64981-310.1007/978-3-642-64981-3Search in Google Scholar
[13] KEGEL, O.: Sylow-gruppen and subnormalteiler endlicher gruppen, Math. Z. 78 (1962), 205–221. http://dx.doi.org/10.1007/BF0119516910.1007/BF01195169Search in Google Scholar
[14] LI, D.— GUO, X.: The influence of c-normality of subgroups on the structure of finite groups II, Comm. Algebra 26 (1998), 1913–1922. http://dx.doi.org/10.1080/0092787980882634210.1080/00927879808826342Search in Google Scholar
[15] LI, Y.— WANG, Y.— WEI, H.: On p-nilpotent of finite groups with some subgroups π-quasinormally embedded, Acta Math. Hungar. 108 (2005), 283–298. http://dx.doi.org/10.1007/s10474-005-0225-810.1007/s10474-005-0225-8Search in Google Scholar
[16] RAMDAN, M.: Influence of normality on maximal subgroups of Sylow subgroups of finite groups, Acta Math. Hungar. 73 (1996), 335–342. http://dx.doi.org/10.1007/BF0005290910.1007/BF00052909Search in Google Scholar
[17] ROBINSON, D.: A Course in Theory of Groups, Springer-Verlag, New York-Heidelberg-Berlin, 1982. http://dx.doi.org/10.1007/978-1-4684-0128-810.1007/978-1-4684-0128-8Search in Google Scholar
[18] SHAALAN, A.: The influence of π-quasinormality of some subgroups on the structure of a finite group, Acta Math. Hungar. 56 (1990), 287–293. http://dx.doi.org/10.1007/BF0190384410.1007/BF01903844Search in Google Scholar
[19] SKIBA, A. N.: On weakly s-permutable subgroups of finite groups, J. Alegebra 315, (2007), 192–209. http://dx.doi.org/10.1016/j.jalgebra.2007.04.02510.1016/j.jalgebra.2007.04.025Search in Google Scholar
[20] SRINIVASAN, S.: Two sufficient conditions for supersolvability of finite groups, Israel J. Math. 35 (1980), 210–214. http://dx.doi.org/10.1007/BF0276119110.1007/BF02761191Search in Google Scholar
[21] WEI, H.: On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups, Comm. Algebra 29 (2001), 2193–2200. http://dx.doi.org/10.1081/AGB-10000217810.1081/AGB-100002178Search in Google Scholar
[22] WIELANDT, H.: Subnormal subgroups and permutation groups Lectures given at Ohio State University, Columbus, Ohio, 1971. Search in Google Scholar
[23] WANG, Y.: C-normality of groups and its properties, J. Algebra 180 (1996), 954–965. http://dx.doi.org/10.1006/jabr.1996.010310.1006/jabr.1996.0103Search in Google Scholar
[24] ZHANG, Q.— WANG, L.: The relationship among subgroups related to normality of subgroups in finite groups, J. Math. Study 36 (2003), 412–417. Search in Google Scholar
[25] ZHU, L.— GUO, W.— SHUM, K. P.: Weakly c-normal subgroups of finite groups and their properties, Comm. Algebra 30 (2002), 5505–5512. http://dx.doi.org/10.1081/AGB-12001566610.1081/AGB-120015666Search in Google Scholar
© 2012 Mathematical Institute, Slovak Academy of Sciences
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
