Prove that $G$ is soluble if and only if it has a subnormal series $1 = G_0 \leq G_1 \leq \cdots \leq G_n = G$ such that each factor $G_i/G_{i+1}$ is abelian.
Since $G$ is soluble, then we have derived series $G \geq G^{(1)} \geq \cdots \geq G^{(k)}=1$. Also $G^{(i)}/G^{(i+1)}$ is abelian.
How should I continue with the proof to show that the series is subnormal? And also the other way of proof?