Lobachevskii Journal of Mathematics

, Volume 39, Issue 1, pp 93–96 | Cite as

Some Properties of Elements of the Group F/[N,N]

  • A. F. Krasnikov


Let F be a free group with basis {x j |jJ}; N a normal subgroup of F. For a given element n of N we describe an elements D l (n), where D l : Z(F) → Z(F) (lJ) are the Fox derivations of the group ring Z(F). If r1, r2 are an elements of F/[N,N] and, for some positive integer d, r1 d is in the normal closure of r2 d in F/[N,N], then r1 is in the normal closure of r2 in F/[N,N]. Let F/N be a soluble group; r an element of F, R the normal closure of r in F. If, for some positive integer k, rN(k) and F/RN(k) is torsion free then F/RN(k+1) is torsion free.

Keywords and phrases

Group ring solvable group 


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  1. 1.
    R. H. Fox, “Free differential calculus. I,” Ann. Math. 57, 517–559 (1953).MathSciNetCrossRefGoogle Scholar
  2. 2.
    W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory (Interscience, New York, 1966).MATHGoogle Scholar
  3. 3.
    W. Magnus, “On a theorem of Marshall Hall,” Ann. Math. 40, 764–768 (1939).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    W. Magnus, “Untersuchungen über einige unendliche diskontinuierliche Gruppen,” Math. Ann. 105, 52–74 (1931).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    P. H. Kropholler, P. A. Linnell, and J. A. Moody, “Applications of a new K-theoretic theorem to soluble group rings,” Proc. Am. Math. Soc. 104, 675–684 (1988).MathSciNetMATHGoogle Scholar
  6. 6.
    N. S. Romanovskii, “Some algorithmic problems for solvable groups,” Algebra Logika 13 (1), 26–34 (1974).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Omsk State UniversityOmskRussia

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