Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

IMPACT FACTOR 2017: 0.831
5-year IMPACT FACTOR: 0.836

CiteScore 2017: 0.68

SCImago Journal Rank (SJR) 2017: 0.450
Source Normalized Impact per Paper (SNIP) 2017: 0.829

Mathematical Citation Quotient (MCQ) 2017: 0.32

ICV 2017: 161.82

Open Access
See all formats and pricing
More options …
Volume 11, Issue 12


Volume 13 (2015)

Groups with every subgroup ascendant-by-finite

Sergio Camp-Mora
  • Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2013-10-08 | DOI: https://doi.org/10.2478/s11533-013-0312-y


A subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.

MSC: 20F19; 20F22; 20F50

Keywords: Ascendant subgroup; Locally nilpotent; Radical; Locally finite group

  • [1] Baer R., Situation der Untergruppen und Struktur der Gruppe, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl., 1933, 2, 12–17 Google Scholar

  • [2] Buckley J.T., Lennox J.C., Neumann B.H., Smith H., Wiegold J., Groups with all subgroups normal-by-finite, J. Austral. Math. Soc., 1995, 59(3), 384–398 http://dx.doi.org/10.1017/S1446788700037289CrossrefGoogle Scholar

  • [3] Dedekind R., Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind, Math. Ann., 1897, 48(4), 548–561 http://dx.doi.org/10.1007/BF01447922CrossrefGoogle Scholar

  • [4] De Falco M., de Giovanni F., Musella C., Group in which every subgroup is permutable-by-finite, Comm. Algebra, 2004, 32(3), 1007–1017 http://dx.doi.org/10.1081/AGB-120027964CrossrefGoogle Scholar

  • [5] De Falco M., de Giovanni F., Musella C., Sysak Y.P., The structure of groups whose subgroups are permutable-byfinite, J. Austral. Math. Soc., 2006, 81(1s), 35–47 http://dx.doi.org/10.1017/S1446788700014622CrossrefGoogle Scholar

  • [6] Dixon M.R., Sylow Theory, Formations and Fitting Classes in Locally Finite Groups, Ser. Algebra, 2, World Scientific, River Edge, 1994 Google Scholar

  • [7] Dixon M.R., Subbotin I.Ya., Groups with finiteness conditions on some subgroup systems: a contemporary stage, Algebra Discrete Math., 2009, 4, 29–54 Google Scholar

  • [8] Lennox J.C., Robinson D.J.S., The Theory of Infinite Soluble Groups, Oxford Math. Monogr., Oxford University Press, Oxford, 2004 http://dx.doi.org/10.1093/acprof:oso/9780198507284.001.0001CrossrefGoogle Scholar

  • [9] Lennox J.C., Stonehewer S.E., Subnormal Subgroups of Groups, Oxford Math. Monogr., Oxford University Press, New York, 1987 Google Scholar

  • [10] Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups, 1&2, Ergeb. Math. Grenzgeb., 62&63, Springer, Berlin-New York, 1972 Google Scholar

  • [11] Schmidt O.Yu., Groups whose all subgroups are special, Mat. Sb., 1924, 31(3–4), 366–372 (in Russian) Google Scholar

  • [12] Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16 http://dx.doi.org/10.1007/BF01111111CrossrefGoogle Scholar

About the article

Published Online: 2013-10-08

Published in Print: 2013-12-01

Citation Information: Open Mathematics, Volume 11, Issue 12, Pages 2182–2185, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-013-0312-y.

Export Citation

© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in