Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The residual nilpotency of the augmentation ideal and the residual nilpotency of some classes of groups


The following theorem is proved: LetG be any group. Then the augmentation ideal ofZG is residually nilpotent if and only ifG is approximated by nilpotent groups without torsion or discriminated by nilpotent pi,-groups,iI, of finite exponents. This theorem is applied to obtain conditions under which the groupsF/N′ are residually nilpotent whereF is a free non-cyclic group and N◃F.

This is a preview of subscription content, log in to check access.


  1. 1.

    M. Auslander and R. C. Lyndon,Commutator subgroups of free groups, Amer. J. Math77 (1955), 929–931.

  2. 2.

    G. Baumslag,Wreath products and extensions, Math. Z.81 (1963), 286–299.

  3. 3.

    A. A. Bovdi,On the intersection of powers of the augmentation ideal, Math. Notes.21 (1967) (Russian).

  4. 4.

    D. Gorenstein,Finite Groups, Harper and Row, 1968.

  5. 5.

    K. Gruenberg,The residual nilpotency of certain presentations of finite groups, Arch. Math.13 (1963), 408–417.

  6. 6.

    K. W. Gruenberg,Some cohomological topics in group theory, Lecture Notes in Math, No. 143, Springer-Verlag, Berlin, 1970.

  7. 7.

    M. Hall,The Theory of Groups, New York, 1959.

  8. 8.

    B. Hartley,The residual nilpotency of wreath products, Proc. London Math. Soc. (3)20 (1970), 365–392.

  9. 9.

    A. Kurosh,The theory of groups, Moscow, 1967 (Russian).

  10. 10.

    A. I. Malcev,Generalized nilpotent algebras and their associated groups, Mat. Sb.25 (67) (1949), 347–366. (In Russian)

  11. 11.

    A. I. Malcev,Nilpotent groups without torsion, Izv. Akad. Nauk SSSR Ser. Mat.13 (1949), 201–212. (In Russian)

  12. 12.

    J. N. Mital,On residual nilpotency, J. London Math. Soc.2 (1970), 337–345.

  13. 13.

    H. Neumann.Varieties of Groups, Springer-Verlag, 1967.

  14. 14.

    I. Passi,Annihilators of relation modules — II, J. Pure Applied Math.6 (1975), 235–237.

  15. 15.

    D. Passman,Advances in group rings, Israel J. Math.19 (1974), 67–108.

Download references

Author information

Correspondence to A. I. Lichtman.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lichtman, A.I. The residual nilpotency of the augmentation ideal and the residual nilpotency of some classes of groups. Israel J. Math. 26, 276–293 (1977).

Download citation


  • Normal Subgroup
  • Nilpotent Group
  • Prime Divisor
  • Wreath Product
  • Finite Order