Transitivity-Systems of Certain One-Relator Groups

  • A. M. Brunner
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 372)


For different non-negative integers n the pairs \(\left( {{a^{{2^n}}},b} \right)\), which are generating pairs of G = gp(a, b; b −1 a 2 b = a 3), are shown to lie in different T-systems of G. The presentation \(\left( {x,y;{x^{ - 1}}{{\left[ {x,y} \right]}^2},\left[ {x,{x^{{y^n}}}} \right]} \right)\) is associated with the generating pair \(\left( {{a^{{2^n}}},b} \right)\) of G for each positive integer n. It is shown that a two generator group which is an extension of an abelian group by an infinite cyclic group has only one T-system of generating pairs. This implies that the quotient group of G by its second derived group has only one T-system of generating pairs.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • A. M. Brunner
    • 1
  1. 1.Institute of Advanced StudiesAustralian National UniversityCanberraAustralia

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