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The Nilpotency Class of Finitely Generated Groups of Exponent Four

  • N. D. Gupta
  • M. F. Newman
Part of the Lecture Notes in Mathematics book series (LNM, volume 372)

Abstract

Wright (Theorem 3 of [6]) has shown that for every positive integer n an n-generator group of exponent 4 is nilpotent of class at most 3n − 1. When n is 2 this result is best possible (see, for example, §3 of [4]). On the other hand, if for some positive integer m the nilpotency class of every m-generator group of exponent 4 were 3m − 3, then groups of exponent 4 would all be soluble [3] and there would be a positive integer k such that for every positive integer n every n-generator group of exponent 4 would have nilpotency class at most n + k (Theorem A of [2]). The gap between these two results is intriguingly small. In this paper we report on some further narrowing of this gap which makes the problem of trying to close the gap still more attractive. We prove the following result.

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References

  1. [1]
    A.J. Bayes, J. Kautsky and J.W. Wamsley, “Computation in nilpotent groups (application)”, these Proc.Google Scholar
  2. [2]
    C.C. Edmunds and N.D. Gupta, “On groups of exponent four IV”, Conf. on Group Theory, University of Wisconsin-Parkside, 1972, pp. 57–70 (Lecture Notes in Mathematics, 319. Springer-Verlag, Berlin, Heidelberg, New York, 1973 ).Google Scholar
  3. [3]
    N.D. Gupta and R.B. Quintana, Jr., “On groups of exponent four. III”, Proc. Amer. Math. Soc. 33 (1972), 15–19. MR45#2000.Google Scholar
  4. [4]
    Marshall Hall, Jr., “Notes on groups of exponent four”, Conf. on Group Theory, University of Wisconsin-Parkside, 1972, pp. 91–118 (Lecture Notes in Mathematics, 319. Springer-Verlag, Berlin, Heidelberg, New York, 1973 ).Google Scholar
  5. [5]
    Wilhelm Magnus, Abraham Karrass, Donald Solitar, Combinatorial group theory (Pure and Appl. Math. 13. Interscience [John Wiley d Sons], New York, London, Sydney, 1966). MR34#7617.Google Scholar
  6. [6]
    C.R.B. Wright, “On the nilpotency class of a group of exponent four”, Pacific J. Math. 11 (1961), 387–394. MR23#A927.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • N. D. Gupta
    • 1
    • 2
  • M. F. Newman
    • 1
    • 2
  1. 1.University of ManitobaWinnipegCanada
  2. 2.Australian National UniversityCanberraAustralia

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