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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0312-y,
author = {Sergio Camp-Mora},
title = {Groups with every subgroup ascendant-by-finite},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {2182-2185},
zbl = {1296.20027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0312-y}
}
Sergio Camp-Mora. Groups with every subgroup ascendant-by-finite. Open Mathematics, Tome 11 (2013) pp. 2182-2185. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0312-y/
[1] Baer R., Situation der Untergruppen und Struktur der Gruppe, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl., 1933, 2, 12–17 | Zbl 0007.05301
[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 | Zbl 0853.20023
[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] 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 | Zbl 1095.20015
[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 | Zbl 1116.20019
[6] Dixon M.R., Sylow Theory, Formations and Fitting Classes in Locally Finite Groups, Ser. Algebra, 2, World Scientific, River Edge, 1994
[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 | Zbl 1199.20051
[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 | Zbl 1059.20001
[9] Lennox J.C., Stonehewer S.E., Subnormal Subgroups of Groups, Oxford Math. Monogr., Oxford University Press, New York, 1987 | Zbl 0606.20001
[10] Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups, 1&2, Ergeb. Math. Grenzgeb., 62&63, Springer, Berlin-New York, 1972 | Zbl 0243.20032
[11] Schmidt O.Yu., Groups whose all subgroups are special, Mat. Sb., 1924, 31(3–4), 366–372 (in Russian)
[12] Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16 http://dx.doi.org/10.1007/BF01111111 | Zbl 0219.20021