In more detail, a group $G$ is called solvable if it has a subnormal series whose factor groups are all abelian, that is, if there are subgroups $\{1\}=G_0< G_1<\cdots< G_k=G$ such that $G_{j-1}$ is normal in $G_j$, and $G_j/G_{j-1}$ is an abelian group, for $j=1,2,\dots,k.$