An Upper Bound for the Index of the Second n-Center Subgroup of An n-Abelian Group

Document Type : Research Paper


Hakim Sabzevari University


Let n be a positive integer. A group G is said to be n-abelian, if (xy)n = xnyn, for any x, y ∈ G. In 1979, Fay and Waals introduced the n- potent and the n-center subgroups of a group G, as Gn = ⟨[x, yn]|x, y ∈ G⟩, Zn(G) = {x ∈ G|xyn = ynx, ∀y ∈ G}, respectively. Also, the second n-center subgroup, Zn ∈ (G), is defined by Zn 2 (G)/Zn(G) = Zn(G/Zn(G)). In this paper, we give an upper bound for the index of the second n-center subgroup of any n-abelian group G in terms of the order of n-potent subgroup Gn


  1. J. L. Alperin, A classification of n-abelian groups, Canad. J. Math., 21, 1238–1244 (1969).
  2. R. Baer, Factorization of n-soluble and n-nilpotent groups, Proc. Amer. Math. Soc., 4, 15–29 (1953).
  3. T. H. Fay, G. L. Waals, Some remarks on n-potent and n-abelian groups, J. Indian. Math. Soc., 47, 217–222 (1983).
  4. P. Hall, Finite-by-nilpotent groups, Proc. Camb. Phil. Soc., 52, 611–616 (1956).
  5. I.M. Isaacs, Derived subgroups and centers of capable groups, Proc. Amer. Math. Soc., 129, 2853–2859 (2001).
  6. F.W. Levi, Notes on group theory I, J. Indian. Math. Soc., 8, 1–7 (1944).
  7. F.W. Levi, Notes on group theory VII, J. Indian. Math. Soc., 9, 37–42 (1945).
  8. I.D. Macdonald, Some explicit bounds in groups with finite derived groups, proc. London math. Soc., 11, 23–56 (1961).
  9. B.H. Neumann, Groups with finite classes of conjugate elements. Proc. London. Math. Soc., 3, 178–187 (1951).
  10. K. Podoski, B. Szegedy, Bounds for the index of the centre in capable groups, Proc. Amer. Math. Soc., 133, 3441–3445 (2005).
  11. K. Podoski, B. Szegedy, Bounds in groups with finite Abelian coverings or with finite derived groups, J. Group Theory, 5, 443fi452 (2002).
  12. M.R. Rismanchian, Some properties of n-capable and n-perfect groups, J. Sci. I.R. Iran, 24, 361–364 (2013).
  13. D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York (1982).
  14. I. Schur, U¨ber die darstellung der endlichen gruppen durch gebrochene lineare substitutionen, Fu¨r. Math. J., 127, 20–50 (1904).