The rank of the factor-group modulo the hypercenter and the rank of the some hypocenter of a group

Leonid Kurdachenko 1  and Javier Otal 2
  • 1 Department of Algebra, National University of Dnepropetrovsk, 72 Gagarin Av., 49010, Dnepropetrovsk, Ukraine
  • 2 Department of Mathematics — IUMA, University of Zaragoza, Pedro Cerbuna 12, 50009, Zaragoza, Spain


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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [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

  • [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

  • [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

  • [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

  • [7] Kurdachenko L.A., Shumyatsky P., The ranks of central factor and commutator groups, Math. Proc. Cambridge Phil. Soc., 2013, 154(1), 63–69

  • [8] Lubotzky A., Mann A., Powerful p-groups. I. Finite groups, J. Algebra, 1987, 105(2), 484–505

  • [9] Makarenko N.Yu., Rank analogues of Hall’s and Baer’s theorems, Siberian Math. J., 2000, 41(6), 1137–1140

  • [10] Schur J., Über die Darstellungen der endlichen Gruppen durch gebrochene lineare substitutionen, J. Reine Angew. Math., 1904, 127, 20–50

  • [11] Wiegold J., Groups with boundedly finite classes of conjugate elements, Proc. Roy. Soc. London Ser. A, 1956, 238(1214), 389–401

  • [12] Wiegold J., Multiplicators and groups with finite central factor-groups, Math. Z., 1965, 89(4), 345–347


Journal + Issues