[[!redirects divisible groups]] ## Definition ## An [[abelian group]] $G$ is a **divisible group** if there exists a left $\mathbb{Z}_{+}$-action $(-)(-):\mathbb{Z}_{+} \times G \to G$, where $\mathbb{Z}_{+}$ is the positve integers, such that for all $n:\mathbb{Z}_{+}$ and all $g:G$, the [[fiber]] of $n(-)$ at $g$ is [[contractible]]: $$\prod_{n:\mathbb{Z}_{+}} \prod_{g:G} \mathrm{isContr}(\mathrm{fiber}(n(-),g))$$ ## See also ## * [[abelian group]] * [[rational numbers]] * [[torsion-free divisible group]] ## References ## * Phillip A. Griffith (1970), Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7