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2009 | 7 | 2 | 186-199

Tytuł artykułu

Groups with small deviation for non-subnormal subgroups

Treść / Zawartość

Warianty tytułu

Języki publikacji

EN

Abstrakty

EN
We introduce the notion of the non-subnormal deviation of a group G. If the deviation is 0 then G satisfies the minimal condition for nonsubnormal subgroups, while if the deviation is at most 1 then G satisfies the so-called weak minimal condition for such subgroups (though the converse does not hold). Here we present some results on groups G that are either soluble or locally nilpotent and that have deviation at most 1. For example, a torsion-free locally nilpotent with deviation at most 1 is nilpotent, while a Baer group with deviation at most 1 has all of its subgroups subnormal.

Słowa kluczowe

Wydawca

Czasopismo

Rocznik

Tom

7

Numer

2

Strony

186-199

Opis fizyczny

Daty

wydano
2009-06-01
online
2009-05-24

Twórcy

  • Algebra Department, Dnepropetrovsk University, Dnepropetrovsk, Ukraine
autor
  • Department of Mathematics, Bucknell University, Lewisburg, USA

Bibliografia

  • [1] Brookes C.J.B., Groups with every subgroup subnormal, Bull. London Math. Soc., 1983, 15, 235–238 http://dx.doi.org/10.1112/blms/15.3.235[Crossref]
  • [2] Evans M.J., Kim Y., On groups in which every subgroup of infinite rank is subnormal of bounded defect, Comm. Algebra, 2004, 32, 2547–2557 http://dx.doi.org/10.1081/AGB-120037398[Crossref]
  • [3] Franciosi S., de Giovanni F., Groups satisfying the minimal condition on non-subnormal subgroups, In: Infinite Groups 1994, de Gruyter, Berlin, 1995, 63–72
  • [4] Hall P., The Edmonton notes on nilpotent groups, Mathematics Department, Queen Mary College, London, 1969
  • [5] Hartley B., McDougall D., Injective modules and soluble groups satisfying the minimal condition for normal subgroups, Bull. Austral. Math. Soc., 1971, 4, 113–135 http://dx.doi.org/10.1017/S0004972700046335[Crossref]
  • [6] Heineken H., Mohamed I.J., A group with trivial centre satisfying the normalizer condition, J. Algebra, 1968, 10, 368–376 http://dx.doi.org/10.1016/0021-8693(68)90086-0[Crossref]
  • [7] Kegel O.H., Wehfritz BAR, Locally finite groups, North-Holland, 1973
  • [8] Kurdachenko L.A., Otal J., Subbotin I.Ya., Groups with prescribed quotient groups and associated module theory, World Scientific Publishing Co., Inc., River Edge, NJ, 2002
  • [9] Kurdachenko L.A., Smith H., Groups with the weak minimal condition for non-subnormal subgroups, Ann. Mat. Pura Appl., 1997, 173, 299–312 http://dx.doi.org/10.1007/BF01783473[Crossref]
  • [10] Kurdachenko L.A., Smith H., Groups in which all subgroups of infinite rank are subnormal, Glasg. Math. J., 2004, 46, 83–89 http://dx.doi.org/10.1017/S0017089503001551[Crossref]
  • [11] Kurdachenko L.A., Smith H., Groups with all subgroups either subnormal or self-normalizing, J. Pure Appl. Algebra, 2005, 196, 271–278 http://dx.doi.org/10.1016/j.jpaa.2004.08.005[Crossref]
  • [12] Lennox J.C., Stonehewer S.E., Subnormal subgroups of groups, The Clarendon Press, Oxford University Press, New York, 1987
  • [13] Matlis E., Cotorsion modules, Mem. Amer. Math. Soc., 1964, 49
  • [14] McConnell J.C., Robson J.C., Noncommutative Noetherian rings, John Wiley & Sons, Ltd., Chichester, 1987
  • [15] Möhres W., Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind, Archiv der Math., 1990, 54, 232–235 http://dx.doi.org/10.1007/BF01188516[Crossref]
  • [16] Robinson D.J.S., Finiteness conditions and generalized soluble groups, Springer-Verlag, New York-Berlin, 1972
  • [17] Robinson D.J.S., A new treatment of soluble groups with finiteness conditions on their abelian subgroups, Bull. London Math. Soc., 1976, 8, 113–129 http://dx.doi.org/10.1112/blms/8.2.113[Crossref]
  • [18] Roseblade J.E., On groups in which every subgroup is subnormal, J. Algebra, 1965, 2, 402–412 http://dx.doi.org/10.1016/0021-8693(65)90002-5[Crossref]
  • [19] Smith H., Groups with few non-nilpotent subgroups, Glasgow Math. J., 1997, 39, 141–151 http://dx.doi.org/10.1017/S0017089500032031[Crossref]
  • [20] Smith H., Torsion-free groups with all subgroups subnormal, Arch. Math., 2001, 76, 1–6 http://dx.doi.org/10.1007/s000130050533[Crossref]
  • [21] Zaitsev D.I., Locally solvable groups of finite rank, Dokl. Akad. Nauk SSSR, 1978, 240, 257–260

Typ dokumentu

Bibliografia

Identyfikatory

Identyfikator YADDA

bwmeta1.element.doi-10_2478_s11533-009-0012-9
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