Structure of n-Abelian groups

  • L. A. Kaluzhnin


Let n be an arbitrary fixed integer. The variety\(\mathfrak{M}\) of n-abelian groups consists of the groups in which the identity (xy)n=xnyn is satisfied. In this paper we show that the variety\(\mathfrak{M}\) is the smallest group variety containing the following three varieties: 1) the variety of abelian groups; 2) the variety defined by the identity xn=1; and 3) the variety defined by the identity xn−1=1. Bibliography of 3 references.


Small Group Abelian Group Group Variety Fixed Integer 
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Literature cited

  1. 1.
    R. Baer, Factorization of n-soluble and n-nilpotent groups, Proc. Amer. Math. Soc.,4, 15–26 (1953).Google Scholar
  2. 2.
    F. Levi, Notes on group theory. I, VII, J. Indian Math. Soc.,8, 1–7 (1944);9, 37–42 (1945).Google Scholar
  3. 3.
    A. L. Shmel'kin, “Interlacings and group varieties,” Izv. AN SSSR. Ser. matem.,29, 149–170 (1965).Google Scholar

Copyright information

© Consultants Bureau 1968

Authors and Affiliations

  • L. A. Kaluzhnin
    • 1
  1. 1.T. G. Shevchenko Kiev State UniversityUSSR

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