Geodesic
Irreducible Algebraic Sets in Metabelian Groups
Algebra i logika, Tome 44 (2005) no. 5, pp. 601-621.

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We present the construction for a $u$-product $G_1\circ G_2$ of two $u$-groups $G_1$ and $G_2$, and prove that $G_1\circ G_2$ is also a $u$-group and that every $u$-group, which contains $G_1$ and $G_2$ as subgroups and is generated by these, is a homomorphic image of $G_1\circ G_2$. It is stated that if $G$ is a $u$-group then the coordinate group of an affine space $G^n$ is equal to $G \circ F_n$, where $F_n$ is a free metabelian group of rank $n$. Irreducible algebraic sets in $G$ are treated for the case where $G$ is a free metabelian group or wreath product of two free Abelian groups of finite ranks.
Mots-clés : $u$-group
Keywords: $u$-product, coordinate group of an affine space, free metabelian group, free Abelian group. 
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     title = {Irreducible {Algebraic} {Sets} in {Metabelian} {Groups}},
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V. N. Remeslennikov; N. S. Romanovskii. Irreducible Algebraic Sets in Metabelian Groups. Algebra i logika, Tome 44 (2005) no. 5, pp. 601-621. http://geodesic.mathdoc.fr/item/AL_2005_44_5_a4/

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