×

On SS-quasinormal and S-quasinormally embedded subgroups of finite groups. (English. Russian original) Zbl 1317.20022

Math. Notes 95, No. 2, 267-276 (2014); translation from Mat. Zametki 95, No. 2, 300-311 (2014).
Summary: A subgroup \(H\) of a group \(G\) is said to be an SS-quasinormal (Supplement-Sylow-quasinormal) subgroup if there is a subgroup \(B\) of \(G\) such that \(HB=G\) and \(H\) permutes with every Sylow subgroup of \(B\). A subgroup \(H\) of a group \(G\) is said to be S-quasinormally embedded in \(G\) if for every Sylow subgroup \(P\) of \(H\), there is an S-quasinormal subgroup \(K\) in \(G\) such that \(P\) is also a Sylow subgroup of \(K\). Groups with certain SS-quasinormal or S-quasinormally embedded subgroups of prime power order are studied.

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] K. Doerk and T. Hawkes, Finite Soluble Groups, in de Gruyter Exp. Math. (Walter de Gruyter, Berlin, 1992), Vol. 4. · Zbl 0753.20001
[2] D. Gorenstein, Finite Groups, in Harper’s Ser. in Modern Math. (Harper &Row Publ., New York, 1968).
[3] B. Huppert, Endliche Gruppen. I, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1967), Vol. 134. · Zbl 0217.07201
[4] L. A. Shemetkov, Formations of Finite Groups, in Contemporary Algebra (Nauka, Moscow, 1978) [in Russian]. · Zbl 0496.20014
[5] M. Assad and A. A. Heliel, ”On S-quasinormally embedded subgroups of finite groups,” J. Pure Appl. Algebra 165(2), 129–135 (2001). · Zbl 1011.20019 · doi:10.1016/S0022-4049(00)00183-3
[6] A. Ballester-Bolinches and M. C. Pedraza-Aguilera, ”Sufficient conditions for supersolubility of finite groups,” J. Pure Appl. Algebra 127(2), 113–118 (1998). · Zbl 0928.20020 · doi:10.1016/S0022-4049(96)00172-7
[7] W. Guo, A. N. Skiba, and K. P. Shum, ”X-quasinormal subgroups,” Sibirsk. Mat. Zh. 48(4), 742–759 (2007) [SiberianMath. J. 48 (4), 593–605 (2007)]. · Zbl 1153.20304
[8] W. Guo, K. P. Shum, and A. N. Skiba, ”G-covering systems of subgroups for classes of p-supersolvable and p-nilpotent finite groups,” Sibirsk. Mat. Zh. 45(3), 527–539 (2004) [Siberian Math. J. 45(3), 433–442 (2004)]. · Zbl 1079.20024
[9] W. Guo, E. V. Legchekova, and A. N. Skiba, ”Finite groups in which every 3-maximal subgroup commutes with all maximal subgroups,” Mat. Zametki 86(3), 350–359 (2009) [Math. Notes 86(3), 325–332 (2009)]. · Zbl 1185.20022 · doi:10.4213/mzm8499
[10] O. H. Kegel, ”Sylow-Gruppen und Subnormalteiler endlicher Gruppen,” Math. Z. 78, 205–221 (1962). · Zbl 0102.26802 · doi:10.1007/BF01195169
[11] Shirong Li, Zhencai Shen, Jianjun Liu, and Xiaochun Liu, ”The influence of SS-quasinormality of some subgroups on the structure of finite groups,” J. Algebra 319(10), 4275–4287 (2008). · Zbl 1152.20019 · doi:10.1016/j.jalgebra.2008.01.030
[12] Shirong Li, Zhencai Shen, and Xianghong Kong, ”On SS-quasinormal subgroups of finite groups,” Comm. Algebra 36(12), 4436–4447 (2008). · Zbl 1163.20011 · doi:10.1080/00927870802179537
[13] L. Miao, ”Finite groups with some maximal subgroups of Sylow subgroups M-supplemented,” Mat. Zametki 86(5), 692–704 (2009) [Math. Notes 86(5), 655–664 (2009)]. · doi:10.4213/mzm8513
[14] Yangming Li and Yanming Wang, ”On S-quasinormally embedded subgroups of finite group,” J. Algebra 281(1), 109–123 (2004). · Zbl 1079.20026 · doi:10.1016/j.jalgebra.2004.06.026
[15] M. Ramadan, ”The influence of S-quasinormality of some subgroups of prime power order on the structure of finite groups,” Arch. Math. (Basel) 77(2), 143–148 (2001). · Zbl 0993.20012 · doi:10.1007/PL00000473
[16] Zhencai Shen, Shirong Li, and Wujie Shi, ”Finite groups with normally embedded subgroups,” J. Group Theory 13(2), 257–265 (2010). · Zbl 1196.20022 · doi:10.1515/jgt.2010.042
[17] Zhencai Shen, Wujie Shi, and Qingliang Zhang, ”S-quasinormality of finite groups,” Front. Math. China 5(2), 329–339 (2010). · Zbl 1200.20014 · doi:10.1007/s11464-010-0010-z
[18] Zhencai Shen and Wujie Shi, ”The influence of SNS-permutability of some subgroups on the structure of finite groups,” Publ. Math. Debrecen 78(1), 159–168 (2011). · Zbl 1260.20034 · doi:10.5486/PMD.2011.4664
[19] L. A. Shemetkov and A.N. Skiba, ”On the X -hypercentre of finite groups,” J. Algebra 322(6), 2106–2117 (2009). · Zbl 1184.20019 · doi:10.1016/j.jalgebra.2009.03.029
[20] A. N. Skiba, ”On weakly s-permutable subgroups of finite groups,” J. Algebra 315(1), 192–209 (2007). · Zbl 1130.20019 · doi:10.1016/j.jalgebra.2007.04.025
[21] A. N. Skiba and O. V. Titov, ”Finite groups with C-quasinormal subgroups,” Sibirsk. Mat. Zh. 48(3), 674–688 (2007) [SiberianMath. J. 48(3), 544–554 (2007)]. · Zbl 1153.20022
[22] A. N. Skiba, ”A note on c-normal subgroups of finite groups,” Algebra Discrete Math., No. 3, 85–95 (2005). · Zbl 1092.20018
[23] Xiaolan Yi and L. A. Shemetkov, ”The formation of finite groups with a supersolvable {\(\pi\)}-Hall subgroup,” Mat. Zametki 87(2), 280–286 (2010) [Math. Notes 87(2), 258–263 (2010)]. · Zbl 1209.20015 · doi:10.4213/mzm7701
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.