Divisible group

Remarks

Example: $\Q$ is divisible: given $a \in \Q$ and $n \ge 1$, take $x = a/n$. Example: $\Q/\Z$ is divisible, as a quotient of the divisible group $\Q$.

Non-example: $\Z$ is not divisible, since there is no integer $x$ with $2x = 1$. A nonzero finite abelian group is not divisible either: if $\abs{A} = m$ and $a \neq 0$, then $mx = 0 \neq a$ for every $x$.

An abelian group is divisible if and only if it is an injective $\Z$-module. It is enough to check the condition for $n$ prime. Every divisible group is isomorphic to a direct sum of copies of $\Q$ and of the Prüfer groups $\Z(p^\infty)$, for various primes $p$.