[[!redirects divisible groups]] ## Definition ## An [[abelian group]] $G$ is a **divisible group** if there exists a multiplicative $\mathbb{Z}_{+}$-action $\alpha:\mathbb{Z}_{+} \to (G \to G)$, where $\mathbb{Z}_{+}$ is the positive integers, such that for all $n:\mathbb{Z}_{+}$ and all $g:G$, the [[fiber]] of $\alpha(n)$ at $g$ is [[contractible]]: $$\prod_{n:\mathbb{Z}_{+}} \prod_{g:G} \mathrm{isContr}(\mathrm{fiber}(\alpha(n),g))$$ ## 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 category: not redirected to nlab yet