Subnormal, permutable, and embedded subgroups in finite groups

James Beidleman; Mathew Ragland

Open Mathematics (2011)

  • Volume: 9, Issue: 4, page 915-921
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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James Beidleman, and Mathew Ragland. "Subnormal, permutable, and embedded subgroups in finite groups." Open Mathematics 9.4 (2011): 915-921. <http://eudml.org/doc/269184>.

@article{JamesBeidleman2011,
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.},
author = {James Beidleman, Mathew Ragland},
journal = {Open Mathematics},
keywords = {Solvable group; PST-group; Subnormal subgroup; S-semipermutable; Seminormal subgroup; finite solvable groups; permutable subgroups; subnormal subgroups; semipermutable subgroups; seminormal subgroups; PST-groups; Sylow subgroups},
language = {eng},
number = {4},
pages = {915-921},
title = {Subnormal, permutable, and embedded subgroups in finite groups},
url = {http://eudml.org/doc/269184},
volume = {9},
year = {2011},
}

TY - JOUR
AU - James Beidleman
AU - Mathew Ragland
TI - Subnormal, permutable, and embedded subgroups in finite groups
JO - Open Mathematics
PY - 2011
VL - 9
IS - 4
SP - 915
EP - 921
AB - 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.
LA - eng
KW - Solvable group; PST-group; Subnormal subgroup; S-semipermutable; Seminormal subgroup; finite solvable groups; permutable subgroups; subnormal subgroups; semipermutable subgroups; seminormal subgroups; PST-groups; Sylow subgroups
UR - http://eudml.org/doc/269184
ER -

References

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  1. [1] Agrawal R.K., Finite groups whose subnormal subgroups permute with all Sylow subgroups, Proc. Amer. Math. Soc., 1975, 47(1), 77–83 http://dx.doi.org/10.1090/S0002-9939-1975-0364444-4 Zbl0299.20014
  2. [2] 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 http://dx.doi.org/10.1515/FORUM.2010.045 Zbl1205.20025
  3. [3] 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 http://dx.doi.org/10.1142/S0219199710003798 Zbl1197.20017
  4. [4] Ballester-Bolinches A., Esteban-Romero R., Sylow permutable subnormal subgroups of finite groups II, Bull. Austr. Math. Soc, 2001, 64(3), 479–486 http://dx.doi.org/10.1017/S0004972700019948 Zbl0999.20012
  5. [5] Ballester-Bolinches A., Esteban-Romero R., Sylow permutable subnormal subgroups of finite groups, J. Algebra, 2002, 251(2), 727–738 http://dx.doi.org/10.1006/jabr.2001.9138 Zbl0999.20012
  6. [6] 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 http://dx.doi.org/10.1515/jgth.2003.010 Zbl1045.20012
  7. [7] 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 Zbl1147.20301
  8. [8] 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 http://dx.doi.org/10.1016/j.jalgebra.2009.01.007 Zbl1190.20015
  9. [9] Beidleman J.C, Ragland M.F., The intersection map of subgroups and certain classes of finite groups, Ric. Mat., 2007, 56(2), 217–227 http://dx.doi.org/10.1007/s11587-007-0015-4 Zbl1167.20014
  10. [10] Kegel O.H., Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z., 1962, 78, 205–221 http://dx.doi.org/10.1007/BF01195169 Zbl0102.26802
  11. [11] Maier R., Zur Vertauschbarkeit und Subnormalität von Untergruppen, Arch. Math. (Basel), 1989, 53(2), 110–120 
  12. [12] Ore O., Contributions to the theory of groups of finite order, Duke Math. J., 1939, 5(2), 431–460 http://dx.doi.org/10.1215/S0012-7094-39-00537-5 Zbl65.0065.06
  13. [13] Robinson D.J.S., A note on finite groups in which normality is transitive, Proc. Amer. Math. Soc., 1968, 19(4), 933–937 http://dx.doi.org/10.1090/S0002-9939-1968-0230808-9 Zbl0159.31002
  14. [14] Schmid P., Subgroups permutable with all Sylow subgroups, J. Algebra, 1998, 207(1), 285–293 http://dx.doi.org/10.1006/jabr.1998.7429 
  15. [15] Wang L, Li Y., Wang Y, Finite groups in which (S-)semipermutability is a transitive relation, Int. J. Algebra, 2008, 2(3) 143–152 Zbl1181.20024
  16. [16] 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 Zbl0136.28302
  17. [17] Zhang Q., s-semipermutability and abnormality in finite groups, Comm. Algebra, 1999, 27(9), 4515–4524 http://dx.doi.org/10.1080/00927879908826711 Zbl0967.20012
  18. [18] 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) Zbl1119.20026

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