Groups whose all subgroups are ascendant or self-normalizing

Leonid Kurdachenko; Javier Otal; Alessio Russo; Giovanni Vincenzi

Open Mathematics (2011)

  • Volume: 9, Issue: 2, page 420-432
  • ISSN: 2391-5455

Abstract

top
This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16].

How to cite

top

Leonid Kurdachenko, et al. "Groups whose all subgroups are ascendant or self-normalizing." Open Mathematics 9.2 (2011): 420-432. <http://eudml.org/doc/269092>.

@article{LeonidKurdachenko2011,
abstract = {This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16].},
author = {Leonid Kurdachenko, Javier Otal, Alessio Russo, Giovanni Vincenzi},
journal = {Open Mathematics},
keywords = {Gruenberg group; Baer group; Subnormal subgroup; Ascendant subgroup; Abnormal subgroup; Pronormal subgroup; Self-normalizing subgroup; Permutable subgroup; ascendant subgroups; locally finite groups; permutable subgroups; self-normalizing subgroups; subnormal subgroups},
language = {eng},
number = {2},
pages = {420-432},
title = {Groups whose all subgroups are ascendant or self-normalizing},
url = {http://eudml.org/doc/269092},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Leonid Kurdachenko
AU - Javier Otal
AU - Alessio Russo
AU - Giovanni Vincenzi
TI - Groups whose all subgroups are ascendant or self-normalizing
JO - Open Mathematics
PY - 2011
VL - 9
IS - 2
SP - 420
EP - 432
AB - This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16].
LA - eng
KW - Gruenberg group; Baer group; Subnormal subgroup; Ascendant subgroup; Abnormal subgroup; Pronormal subgroup; Self-normalizing subgroup; Permutable subgroup; ascendant subgroups; locally finite groups; permutable subgroups; self-normalizing subgroups; subnormal subgroups
UR - http://eudml.org/doc/269092
ER -

References

top
  1. [1] Baer R., Situation der Untergruppen und Struktur der Gruppe, Sitzungsber. Heidelb. Akad. Wiss., 1933, 2, 12–17 Zbl0007.05301
  2. [2] Ballester-Bolinches A., Kurdachenko L.A., Otal J., Pedraza T., Infinite groups with many permutable subgroups, Rev. Mat. Iberoam., 2008, 24(3), 745–764 Zbl1175.20036
  3. [3] Belyaev V.V., Groups of Miller-Moreno type, Sibirsk. Math. Zh., 1979, 19(3), 509–514 (in Russian) Zbl0409.20027
  4. [4] Belyaev V.V., Sesekin N.F., Infinite groups of Miller-Moreno type, Acta Acad. Math. Sci. Hungar., 1975, 26(3–4), 369–376 (in Russian) http://dx.doi.org/10.1007/BF01902346 
  5. [5] Casolo C., Torsion-free groups in which every subgroup is subnormal, Rend. Circ. Mat. Palermo, 2001, 50(2), 321–324 http://dx.doi.org/10.1007/BF02844986 Zbl1138.20306
  6. [6] Chernikov S.N., Groups with given properties of systems of infinite subgroups, Ukrainian Math. J., 1967, 19(6), 715–731 http://dx.doi.org/10.1007/BF01105854 
  7. [7] Chernikov S.N., On normalizer condition, Mat. Zametki, 1968, 3(1), 45–50 (in Russian) Zbl0186.32004
  8. [8] De Falco M., Kurdachenko L.A., Subbotin I.Ya., Groups with only abnormal and subnormal subgroups, Atti Sem. Mat. Fis. Univ. Modena, 1988, 46(2), 435–442 
  9. [9] De Falco M., Musella C., A normalizer condition for modular subgroups, In: Advances in Group Theory, Aracne, Rome, 2002, 163–172 Zbl1057.20023
  10. [10] Dedekind R., Über Gruppen, deren sämmtliche Theiler Normaltheiler sind, Math. Ann., 1897, 48(4), 548–561 http://dx.doi.org/10.1007/BF01447922 
  11. [11] Ebert G., Bauman S., A note on subnormal and abnormal chains, J. Algebra, 1975, 36(2), 287–293 http://dx.doi.org/10.1016/0021-8693(75)90103-9 Zbl0314.20019
  12. [12] Fattahi A., Groups with only normal and abnormal subgroups, J. Algebra, 1974, 28(1), 15–19 http://dx.doi.org/10.1016/0021-8693(74)90019-2 Zbl0274.20022
  13. [13] Franciosi S., de Giovanni F., On groups with many subnormal subgroups, Note Mat., 1993, 13(1), 99–105 Zbl0809.20019
  14. [14] Giordano G., Gruppi con normalizzatori estremali, Matematiche (Catania), 1971, 26, 291–296 
  15. [15] Gruenberg K.W., The Engel elements of a soluble group, Illinois J. Math., 1959, 3(2), 151–168 Zbl0092.02102
  16. [16] Heineken H., Kurdachenko L.A., Groups with subnormality for all subgroups that are not finitely generated, Ann. Mat. Pura Appl., 1995, 169, 203–232 http://dx.doi.org/10.1007/BF01759354 Zbl0848.20023
  17. [17] Huppert B., Zur Sylowstruktur auflösbarer Gruppen, Arch. Math. (Basel), 1961, 12, 161–169 
  18. [18] Kurdachenko L.A., Smith H., Groups with all subgroups either subnormal or self-normalizing, J. Pure Appl. Algebra, 2005, 196(2–3), 271–278 http://dx.doi.org/10.1016/j.jpaa.2004.08.005 Zbl1078.20026
  19. [19] Miller G.A., Moreno H., Non-abelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc., 1903, 4(4), 389–404 Zbl34.0173.01
  20. [20] Möhres W., Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind, Arch. Math. (Basel), 1990, 54(3), 232–235 Zbl0663.20027
  21. [21] Plotkin B.I., On the theory of locally nilpotent groups, Dokl. Akad. Nauk SSSR, 1951, 76, 639–641 (in Russian) 
  22. [22] Rose J.S., A Course on Group Theory, Cambridge University Press, Cambridge-New York-Melbourne, 1978 Zbl0371.20001
  23. [23] Roseblade J.E., On groups in which every subgroup is subnormal, J. Algebra, 1965, 2(4), 402–412 http://dx.doi.org/10.1016/0021-8693(65)90002-5 Zbl0135.04901
  24. [24] Schmidt O.Yu., Groups whose all subgroups are special, Mat. Sb., 1924, 31(3–4), 366–372 (in Russian) 
  25. [25] Schmidt O.Yu., On infinite special groups, Mat. Sb., 1940, 8(50)(3), 363–375 (in Russian) Zbl66.0070.01
  26. [26] Schmidt R., Subgroup Lattices of Groups, de Gruyter Exp. Math., 14, Walter de Gruyter, Berlin, 1994 
  27. [27] Shumyatsky P., Locally finite groups with an automorphism whose centralizer is small, In: Topics in Infinite Groups, Quad. Mat., 8, Dept. Math., Seconda Univ. Napoli, Caserta, 2001, 278–296 Zbl1038.20014
  28. [28] Smith H., Torsion-free groups with all subgroups subnormal, Arch. Math. (Basel), 2001, 76(1), 1–6 Zbl0982.20018
  29. [29] Smith H., Torsion-free groups with all non-nilpotent subgroups subnormal, In: Topics in Infinite Groups, Quad. Mat., 8, Dept. Math., Seconda Univ. Napoli, Caserta, 2001, 297–308 Zbl1017.20017
  30. [30] Smith H., Groups with all non-nilpotent subgroups subnormal, In: Topics in Infinite Groups, Quad. Mat., 8, Dept. Math., Seconda Univ. Napoli, Caserta, 2001, 309–326 Zbl1017.20018
  31. [31] Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16 http://dx.doi.org/10.1007/BF01111111 Zbl0219.20021
  32. [32] Tomkinson M.J., FC-Groups, Pitman Res. Notes Math. Ser., 96, Pitman, Boston-London-Melbourne, 1994 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.