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Transitivity-Systems of Certain One-Relator Groups

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Part of the Lecture Notes in Mathematics book series (LNM,volume 372)

Abstract

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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References

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© 1974 Springer-Verlag Berlin Heidelberg

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Brunner, A.M. (1974). Transitivity-Systems of Certain One-Relator Groups. In: Newman, M.F. (eds) Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21571-5_10

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  • DOI: https://doi.org/10.1007/978-3-662-21571-5_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06845-7

  • Online ISBN: 978-3-662-21571-5

  • eBook Packages: Springer Book Archive