Proposition 10

Let G be a group and n an integer. The following conditions are equivalent:

(i) G is solvable of class $\leq n$. (ii) There exists a series of normal subgroups of G

such that the groups $G^k/G^{k+1}$ are commutative. (iii) There exists a series of subgroups of G such that, for all k, $G^{k+1}$ is a normal subgroup of $G^k$ and $G^k/G^{k+1}$ is commutative.

(iv) There exists a normal commutative subgroup $A$ of $G$ such that $G/A$ is solvable of class $\leq n - 1$.

For (i) $\Rightarrow$ (ii) it suffices to take $G^k$ equal to $D^k(G)$. (ii) $\Rightarrow$ (iii) trivially. (iii) $\Rightarrow$ (i) for $D^k(G)$ is necessarily contained in $G^k$. The equivalence of (ii) and (iv) is immediate by induction on $n$.

More briefly: a group is solvable of class $\leq n$ if it can be obtained by successive extensions of $n$ commutative groups.