Abstract
The purpose of this paper is to study the subgroup embedding properties of S-semipermutability, semipermutability, and seminormality. Here we say H is S-semipermutable (resp. semipermutable) in a group Gif H permutes which each Sylow subgroup (resp. subgroup) of G whose order is relatively prime to that of H. We say H is seminormal in a group G if H is normalized by subgroups of G whose order is relatively prime to that of H. In particular, we establish that a seminormal p-subgroup is subnormal. We also establish that the solvable groups in which S-permutability is a transitive relation are precisely the groups in which the subnormal subgroups are all S-semipermutable. Local characterizations of this result are also established.
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References
Agrawal R.K., Finite groups whose subnormal subgroups permute with all Sylow subgroups, Proc. Amer. Math. Soc., 1975, 47(1), 77–83
Al-Sharo K.A., Beidleman J.C., Heineken H., Ragland M.F., Some characterizations of finite groups in which semiper-mutability is a transitive relation, Forum Math., 2010, 22(5), 855–862
Ballester-Bolinches A., Cossey J., Soler-Escrivà X., On a permutability property of subgroups of finite soluble groups, Commun. Contemp. Math., 2010, 12(2), 207–221
Ballester-Bolinches A., Esteban-Romero R., Sylow permutable subnormal subgroups of finite groups II, Bull. Austr. Math. Soc, 2001, 64(3), 479–486
Ballester-Bolinches A., Esteban-Romero R., Sylow permutable subnormal subgroups of finite groups, J. Algebra, 2002, 251(2), 727–738
Beidleman J.C., Heineken H., Finite soluble groups whose subnormal subgroups permute with certain classes of subgroups, J. Group Theory, 2003, 6(2), 139–158
Beidleman J.C, Heineken H., Pronormal and subnormal subgroups and permutability, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 2003, 6(3), 605–615
Beidleman J.C, Heineken H., Ragland M.F., Solvable PST-groups, strong Sylow bases and mutually permutable products, J. Algebra, 2009, 321(7), 2022–2027
Beidleman J.C, Ragland M.F., The intersection map of subgroups and certain classes of finite groups, Ric. Mat., 2007, 56(2), 217–227
Kegel O.H., Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z., 1962, 78, 205–221
Maier R., Zur Vertauschbarkeit und Subnormalität von Untergruppen, Arch. Math. (Basel), 1989, 53(2), 110–120
Ore O., Contributions to the theory of groups of finite order, Duke Math. J., 1939, 5(2), 431–460
Robinson D.J.S., A note on finite groups in which normality is transitive, Proc. Amer. Math. Soc., 1968, 19(4), 933–937
Schmid P., Subgroups permutable with all Sylow subgroups, J. Algebra, 1998, 207(1), 285–293
Wang L, Li Y., Wang Y, Finite groups in which (S-)semipermutability is a transitive relation, Int. J. Algebra, 2008, 2(3) 143–152
Zacher G., I gruppi risolubili finiti in cui i sottogruppi di composizione coincidono con i sottogruppi quasi-normali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 1964, 37, 150–154
Zhang Q., s-semipermutability and abnormality in finite groups, Comm. Algebra, 1999, 27(9), 4515–4524
Zhang Q.H., Wang L.F, The influence of s-semipermutable subgroups on finite groups, Acta Math. Sinica (Chin. Ser.), 2005, 48(1), 81–88 (in Chinese)
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Dedicated to Professor Hermonn Heineken on the occasion of his seventy-fifth birthday.
An erratum to this article can be found at http://dx.doi.org/10.2478/s11533-011-0098-8
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Beidleman, J.C., Ragland, M.F. Subnormal, permutable, and embedded subgroups in finite groups. centr.eur.j.math. 9, 915–921 (2011). https://doi.org/10.2478/s11533-011-0029-8
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DOI: https://doi.org/10.2478/s11533-011-0029-8