We compare the special rank of the factors of the upper central series and terms of the lower central series of a group. As a consequence we are able to show some generalizations of a theorem of Reinhold Baer.
@article{bwmeta1.element.doi-10_2478_s11533-013-0284-y,
author = {Leonid Kurdachenko and Javier Otal},
title = {The rank of the factor-group modulo the hypercenter and the rank of the some hypocenter of a group},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {1732-1741},
zbl = {1316.20039},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0284-y}
}
Leonid Kurdachenko; Javier Otal. The rank of the factor-group modulo the hypercenter and the rank of the some hypocenter of a group. Open Mathematics, Tome 11 (2013) pp. 1732-1741. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0284-y/
[1] Adian S.I., Certain torsion-free groups, Izv. Akad. Nauk SSSR Ser. Mat., 1971, 35, 459–468 (in Russian)
[2] Baer R., Endlichkeitskriterien für Kommutatorgruppen, Math. Ann., 1952, 124, 161–177 http://dx.doi.org/10.1007/BF01343558[Crossref]
[3] Franciosi S., de Giovanni F., Kurdachenko L.A., The Schur property and groups with uniform conjugacy classes, J. Algebra, 1995, 174(3), 823–847 http://dx.doi.org/10.1006/jabr.1995.1155[Crossref] | Zbl 0831.20043
[4] Kaloujnine L.A., Über gewisse Beziehungen zwischen eine Gruppe und ihren Automorphismen, In: Bericht über die Mathematiker-Tagung, Berlin, January 1953, Deutscher Verlag der Wissenschaften, Berlin, 1953, 164–172 | Zbl 0053.01002
[5] Kurdachenko L.A., Artinian Modules over Group Rings, In: Infinite Groups and Related Algebraic Structures, Akad. Nauk Ukrainy, Inst. Mat., Kiev, 1993, 160–177 (in Russian)
[6] Kurdachenko L.A., Otal J., Subbotin I.Ya., Artinian Modules over Group Rings, Front. Math., Birkhäuser, Basel, 2007 | Zbl 1110.16001
[7] Kurdachenko L.A., Shumyatsky P., The ranks of central factor and commutator groups, Math. Proc. Cambridge Phil. Soc., 2013, 154(1), 63–69 http://dx.doi.org/10.1017/S0305004112000412[Crossref] | Zbl 1272.20032
[8] Lubotzky A., Mann A., Powerful p-groups. I. Finite groups, J. Algebra, 1987, 105(2), 484–505 http://dx.doi.org/10.1016/0021-8693(87)90211-0[Crossref] | Zbl 0626.20010
[9] Makarenko N.Yu., Rank analogues of Hall’s and Baer’s theorems, Siberian Math. J., 2000, 41(6), 1137–1140 http://dx.doi.org/10.1023/A:1004828422007[Crossref]
[10] Schur J., Über die Darstellungen der endlichen Gruppen durch gebrochene lineare substitutionen, J. Reine Angew. Math., 1904, 127, 20–50 | Zbl 35.0155.01
[11] Wiegold J., Groups with boundedly finite classes of conjugate elements, Proc. Roy. Soc. London Ser. A, 1956, 238(1214), 389–401 | Zbl 0077.03002
[12] Wiegold J., Multiplicators and groups with finite central factor-groups, Math. Z., 1965, 89(4), 345–347 http://dx.doi.org/10.1007/BF01112166[Crossref] | Zbl 0134.03002