On the order of the n-center factor subgroup of an n-abelian group

Document Type : Research Paper


Hakim Sabzevari University


A group G is said to be n-abelian, if (xy)n=xnyn, for any x,y ∈in G and a positive integer n. In 1979, Fay and Waals introduced the n-potent and the n-center subgroups of a group G, denoted by Gn and Zn(G), respectively. In this paper, we show that the index of the n-center is bounded by an order power of the n-potent subgroup, for some classes of groups. In fact for all n-abelian groups G with finite n-potent subgroup, we prove that if G/Zn(G) is finitely generated, then [G : Zn(G)] ≤ |Gn|d(G/Zn(G)). Moreover, we conclude that [G : Zn(G)] ≤ |Gn|2log2|Gn|, for some n-capable group G.


  1. T. H. Fay, G. L. Waals, Some remarks on n-potent and n-abelian groups, J. Indian. Math. Soc., 47, 217–222 (1983).
  2. P. Hall, Finite-by-nilpotent groups, Proc. Camb. Phil. Soc., 52, 611–616 (1956).
  3. F. W. Levi, Notes on group theory I, J. Indian. Math. Soc., 8, 1–7 (1944).
  4. F. W. Levi, Notes on group theory VII, J. Indian. Math. Soc., 9, 37–42 (1945).
  5. B. H. Neumann, Groups with finite classes of conjugate elements. Proc. London. Math. Soc., 3, 178–187 (1951).
  6. P. Niroomand, The converse of Schur’s theorem, Arch. Math., 94, 401–403 (2010).
  7. K. Podoski, B. Szegedy, Bounds for the index of the centre in capable groups, Proc. Amer. Math. Soc., 133, 3441–3445 (2005).
  8. I. Schur, U¨ber die darstellung der endlichen gruppen durch gebrochene lineare substitutionen, Fu¨r. Math. J., 127, 20–50 (1904).
  9. B. Sury, A generalization of a converse of Schur’s theorem, Arch. Math., 95, 317–318 (2010).
  10. J. Wiegold, Multiplicators and groups with finite central factor-groups, Math. Z., 89, 345–347 (1965).
  11. M. K. Yadav, Converse of Schur’s theorem -A statement, arXiv:1212.2710v2 [math.GR], (2012).
  12. M. K. Yadav, Converse of Schur’s theorem and arguments of B.H. Neumann, arXiv:1011.2083v3 [math.GR], (2015).