Ok, so abelian groups are solvable.
And Thm II.8.5 of Hungerford says A group is solvable iff it has a solvable series. (The group may be finite or infinite.)
However, I can't seem to find a solvable series for $\mathbb{Z}$, for example $\mathbb{Z},2\mathbb{Z},6\mathbb{Z},\ldots$ will not terminate in the identity group.
Someone said that $\mathbb{Z},\left\{0\right\}$ is a solvable series for $\mathbb{Z}$. Is this a definition set by Hungerford since $\mathbb{Z},\left\{0\right\}$ is not even a composition series : $\mathbb{Z}/\left\{0\right\}$ is not a simple group.