Let G be a hypercentral group. Our main result here is that if G/G’ is divisible by finite then G itself is divisible by finite. This extends a recent result of Heng, Duan and Chen [2], who prove in a slightly weaker form the special case where G is also a p-group. If G is torsion-free, then G is actually divisible.
@article{bwmeta1.element.doi-10_2478_s11533-007-0015-3,
author = {B. Wehrfritz},
title = {On hypercentral groups},
journal = {Open Mathematics},
volume = {5},
year = {2007},
pages = {596-606},
zbl = {1133.20021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0015-3}
}
B. Wehrfritz. On hypercentral groups. Open Mathematics, Tome 5 (2007) pp. 596-606. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0015-3/
[1] L. Fuchs: Infinite Abelian Groups, Vol. 1, Academic Press, New York, 1970. | Zbl 0209.05503
[2] L. Heng, Z. Duan and G. Chen: “On hypercentral groups G with |G: G m | < ∞”, Comm. Algebra, Vol. 34, (2006), pp. 1803–1810. http://dx.doi.org/10.1080/00927870500542770 | Zbl 1105.20030
[3] O.H. Kegel and B.A.F. Wehrfritz: Locally Finite Groups, North-Holland, Amsterdam, 1973.
[4] D.H. McLain: “Remarks on the upper central series of a group”, Proc. Glasgow Math. Assoc., Vol. 3, (1956), pp. 38–44. http://dx.doi.org/10.1017/S2040618500033414 | Zbl 0072.25702
[5] D.J.S. Robinson: Finiteness Conditions and Generalized Soluble Groups, Springer-Verlag, Berlin, 1972. | Zbl 0243.20032
[6] B.A.F. Wehrfritz: “Nilpotence in finitary linear groups”, Michigan Math. J., Vol. 40, (1992), pp. 419–432. | Zbl 0807.20040