[[!redirects torsion-free divisible groups]] [[!redirects Q-module]] [[!redirects Q-modules]] [[!redirects Q-vector space]] [[!redirects Q-vector spaces]] ## Definition ## A [[divisible group]] $G$ is **torsion-free** if the only integer $n:\mathbb{Z}$ such that $\alpha(n)(g) = 0$ for all $g:G$ is $0$. ## Properties ## * Just as every abelian group is a $\mathbb{Z}$-[[module]], every torsion-free divisible group is a $\mathbb{Q}$-[[module]], or a $\mathbb{Q}$-[[vector space]]. ## See also ## * [[divisible group]] * [[rational numbers]] * [[rationalization of an abelian group]] * [[rational homotopy type]] * [[Q-algebra]] ## References ## * Phillip A. Griffith (1970), Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7