×

On nearly \(SS\)-embedded subgroups of finite groups. (English) Zbl 1308.20021

Summary: Let \(H\) be a subgroup of a finite group \(G\). \(H\) is nearly \(SS\)-embedded in \(G\) if there exists an \(S\)-quasinormal subgroup \(K\) of \(G\), such that \(HK\) is \(S\)-quasinormal in \(G\) and \(H\cap K\leq H_{\mathrm{se\,}G}\), where \(H_{\mathrm{se\,}G}\) is the subgroup of \(H\), generated by all those subgroups of \(H\) which are \(S\)-quasinormally embedded in \(G\). In this paper, the authors investigate the influence of nearly \(SS\)-embedded 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
20D15 Finite nilpotent groups, \(p\)-groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Assad, M. and Heliel, A. A., On S-quasinormally embedded subgroups of finite groups, J. Pure Appl. Algebra, 165, 2001, 129-135. · Zbl 1011.20019 · doi:10.1016/S0022-4049(00)00183-3
[2] Ballester-Bolinches, A. and Pedraza-Aguilera, M. C., Sufficient conditions for supersolubility of finite groups, J. Pure Appl. Algebra, 127, 1998, 113-118. · Zbl 0928.20020 · doi:10.1016/S0022-4049(96)00172-7
[3] Deskins, W. E., On quasinormal subgroups of finite groups, Math. Z., 82, 1963, 125-132. · Zbl 0114.02004 · doi:10.1007/BF01111801
[4] Doerk, K. and Hawkes, T., Finite Solvable Groups, Walter de Gruyter, New York, 1992. · Zbl 0753.20001 · doi:10.1515/9783110870138
[5] Gorenstein, D., Finite Groups, Harper and Row Publishers, New York, Evanston, London, 1968. · Zbl 0185.05701
[6] Guo, W., The Theory of Class of Groups, Science Press-Kluwer Academic Publishers, Beijing, New York, Dordrecht, Boston, London, 2000.
[7] Guo, W., Lu, Y. and Niu, W., s-embedded subgroups of finite groups, Algebra and Logic, 49(4), 2010, 293-304. · Zbl 1255.20021 · doi:10.1007/s10469-010-9097-2
[8] Guo, W., Shum, K. P. and Skiba, A. N., On solubility and supersolubility of some classes of finite groups, Sci. China Ser. A, 52, 2009, 272-286. · Zbl 1189.20023 · doi:10.1007/s11425-009-0008-8
[9] Guo, W. and Skiba, A. N., Finite groups with given s-embedded and n-embedded subgroups, J. Algebra, 321, 2009, 2843-2860. · Zbl 1182.20026 · doi:10.1016/j.jalgebra.2009.02.016
[10] Guo, W., Skiba, A. N. and Yang, N., SE-supplemented subgroups of finite groups, Rend. Sem. Mat. Univ. Padova, 129, 2013, 245-263. · Zbl 1287.20025 · doi:10.4171/RSMUP/129-14
[11] Guo, X. and Shum, K. P., On c-normal maximal and minimal subgroups of Sylow p-subgroups of finite groups, Arch. Math., 80, 2003, 561-569. · Zbl 1050.20010 · doi:10.1007/s00013-003-0810-4
[12] Huppert, B., Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York, 1967. · Zbl 0217.07201 · doi:10.1007/978-3-642-64981-3
[13] Kegel, O., Sylow-Gruppen and subnormalteiler endlicher gruppen, Math. Z., 78, 1962, 205-221. · Zbl 0102.26802 · doi:10.1007/BF01195169
[14] Li, J., Chen, G. and Chen, R., On weakly s-embedded subgroups of finite groups, Sci. China Math., 54, 2011, 1899-1908. · Zbl 1239.20025 · doi:10.1007/s11425-011-4239-0
[15] Li, S. and Li, Y., On S-quasinormal and c-normal subgroups of a finite group, Czechoslovak Math. J., 58(133), 2008, 1083-1095. · Zbl 1166.20013 · doi:10.1007/s10587-008-0070-3
[16] Miao, L., On weakly s-permutable subgroups of finite groups, Bull. Braz. Math. Soc. New Series, 41(2), 2010, 223-235. · Zbl 1221.20014 · doi:10.1007/s00574-010-0011-2
[17] Robinson, D. J. S., A Course in Theory of Group, Spinger-Verlag, New York, 1982. · doi:10.1007/978-1-4684-0128-8
[18] Schmid, P., Subgroups permutable with all Sylow subgroups, J. Algebra, 207, 1998, 285-293. · Zbl 0910.20015 · doi:10.1006/jabr.1998.7429
[19] Srinivasan, S., Two sufficient conditions for supersolvability of finite groups, Israel Journal of Mathematics, 35, 1980, 210-214. · Zbl 0437.20012 · doi:10.1007/BF02761191
[20] Thompson, J. G., Normal p-complements for finite groups, J. Algebra, 1, 1964, 43-46. · Zbl 0119.26802 · doi:10.1016/0021-8693(64)90006-7
[21] Wang, Y., c-normality of groups and its properties, J. Algebra, 180, 1996, 954-965. · Zbl 0847.20010 · doi:10.1006/jabr.1996.0103
[22] Wang, Y. and Guo, W., Nearly s-normality of groups and its properties, Comm. Algebra, 38, 2010, 3821-3836. · Zbl 1221.20015 · doi:10.1080/00927870903286850
[23] Wielandt, H., Subnormal Subgroups and Permutation Groups, Lectures Given at the Ohio State University, Columbus, Ohio, 1971.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.