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On the number of relations in free products of abelian groups

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Abstract

We consider the finitely generated groups constructed from cyclic groups by free and direct products and study the question of the smallest number of relations for a given system of generators. This question is related to the relation gap problem. We prove that if m and n are not coprime then the group H m,n = (ℤ m × ℤ) * (ℤ n × ℤ) cannot be defined using three relations in the standard system of generators. We obtain a similar result for the groups G m,n = (ℤ m × ℤ m ) * (ℤ n × ℤ n ). On the other hand, we establish that for coprime m and n the image of H m,n in every nilpotent group is defined using three relations.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, Russia

    V. G. Bardakov & M. V. Neshchadim

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  1. V. G. Bardakov
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  2. M. V. Neshchadim
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Correspondence to V. G. Bardakov.

Additional information

Original Russian Text Copyright © 2012 Bardakov V.G. and Neshchadim M.V.

The authors were supported by the Program “Development of the Scientific Potential of Higher School” (Grant 2.1.1.10726), the Federal Target Program “Scientific and Pedagogical Personnel of Innovation Russia” for 2009–2013 (State Contract 02.740.11.5191), and the Russian Foundation for Basic Research (Grant 10-01-00642).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 4, pp. 741-750, July–August, 2012.

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Bardakov, V.G., Neshchadim, M.V. On the number of relations in free products of abelian groups. Sib Math J 53, 591–599 (2012). https://doi.org/10.1134/S0037446612040027

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  • DOI: https://doi.org/10.1134/S0037446612040027

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