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Sibirsk. Mat. Zh., 2012, Volume 53, Number 4, Pages 741–751 (Mi smj2360)  

On the number of relations in free products of abelian groups

V. G. Bardakovab, M. V. Neshchadimba

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University, Novosibirsk

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 $H_{m,n}=(\mathbb Z_m\times\mathbb Z)*(\mathbb Z_n\times\mathbb Z)$ cannot be defined using three relations in the standard system of generators. We obtain a similar result for the groups $G_{m,n}=(\mathbb Z_m\times\mathbb Z_m)*(\mathbb Z_n\times Z_n)$. 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.

Keywords: finitely presented group, minimal number of relations, module of relations, relation gap.

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English version:
Siberian Mathematical Journal, 2012, 53:4, 591–599

Bibliographic databases:

UDC: 512.8
Received: 27.07.2011

Citation: V. G. Bardakov, M. V. Neshchadim, “On the number of relations in free products of abelian groups”, Sibirsk. Mat. Zh., 53:4 (2012), 741–751; Siberian Math. J., 53:4 (2012), 591–599

Citation in format AMSBIB
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\paper On the number of relations in free products of abelian groups
\jour Sibirsk. Mat. Zh.
\yr 2012
\vol 53
\issue 4
\pages 741--751
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\jour Siberian Math. J.
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\pages 591--599
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