Groups with every subgroup ascendant-by-finite

Sergio Camp-Mora

Open Mathematics (2013)

  • Volume: 11, Issue: 12, page 2182-2185
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Sergio Camp-Mora. "Groups with every subgroup ascendant-by-finite." Open Mathematics 11.12 (2013): 2182-2185. <http://eudml.org/doc/269049>.

@article{SergioCamp2013,
abstract = {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.},
author = {Sergio Camp-Mora},
journal = {Open Mathematics},
keywords = {Ascendant subgroup; Locally nilpotent; Radical; Locally finite group; locally finite groups; subgroups of finite index; ascendant subgroups; permutable subgroups; locally nilpotent subgroups},
language = {eng},
number = {12},
pages = {2182-2185},
title = {Groups with every subgroup ascendant-by-finite},
url = {http://eudml.org/doc/269049},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Sergio Camp-Mora
TI - Groups with every subgroup ascendant-by-finite
JO - Open Mathematics
PY - 2013
VL - 11
IS - 12
SP - 2182
EP - 2185
AB - 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.
LA - eng
KW - Ascendant subgroup; Locally nilpotent; Radical; Locally finite group; locally finite groups; subgroups of finite index; ascendant subgroups; permutable subgroups; locally nilpotent subgroups
UR - http://eudml.org/doc/269049
ER -

References

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  1. [1] Baer R., Situation der Untergruppen und Struktur der Gruppe, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl., 1933, 2, 12–17 Zbl0007.05301
  2. [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/S1446788700037289 Zbl0853.20023
  3. [3] Dedekind R., Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind, Math. Ann., 1897, 48(4), 548–561 http://dx.doi.org/10.1007/BF01447922 
  4. [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-120027964 Zbl1095.20015
  5. [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/S1446788700014622 Zbl1116.20019
  6. [6] Dixon M.R., Sylow Theory, Formations and Fitting Classes in Locally Finite Groups, Ser. Algebra, 2, World Scientific, River Edge, 1994 
  7. [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 Zbl1199.20051
  8. [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/9780198507284.001.0001 Zbl1059.20001
  9. [9] Lennox J.C., Stonehewer S.E., Subnormal Subgroups of Groups, Oxford Math. Monogr., Oxford University Press, New York, 1987 Zbl0606.20001
  10. [10] Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups, 1&2, Ergeb. Math. Grenzgeb., 62&63, Springer, Berlin-New York, 1972 Zbl0243.20032
  11. [11] Schmidt O.Yu., Groups whose all subgroups are special, Mat. Sb., 1924, 31(3–4), 366–372 (in Russian) 
  12. [12] Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16 http://dx.doi.org/10.1007/BF01111111 Zbl0219.20021

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