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In this paper we deal with Jordaness properties of the groups $Bir(X)$ of birational automorphisms of irreducible smooth projective varieties $X$ over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov - C. Shramov) that $Bir(X)$ is Jordan if $X$ is non-uniruled. 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Zarhin","submitted_at":"2015-12-06T05:52:38Z","abstract_excerpt":"A group $G$ is called Jordan if there is a positive integer $J=J_G$ such that every finite subgroup $\\mathcal{B}$ of $G$ contains a commutative subgroup $\\mathcal{A}\\subset \\mathcal{B}$ such that $\\mathcal{A}$ is normal in $\\mathcal{B}$ and the index $[\\mathcal{B}:\\mathcal{A}] \\le J$ (V.L. Popov). In this paper we deal with Jordaness properties of the groups $Bir(X)$ of birational automorphisms of irreducible smooth projective varieties $X$ over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov - C. Shramov) that $Bir(X)$ is Jordan if $X$ is non-uniruled. 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