Definition 7

Let G be a group. The derived series of G is the series (D^n(G)){n \in \mathbf{N}} defined inductively by:*

Then $D^0(G) = C^1(G) = G,\ D^1(G) = C^2(G) = D(G) = (G, G)$. For all n \in \mathbf{N}, D^n(G) \subset C^{2^n}(G), as is seen by induction on n using formula (7) of no. 3.

Let $f : G \to G'$ be a group homomorphism. It is seen, by induction on n, that $f(D^n(G)) \subset D^n(G')$ and that, if $f$ is surjective, $f(D^n(G)) = D^n(G')$. In particular, for all $n \in \mathbf{N}$, $D^n(G)$ is a characteristic (and therefore normal) subgroup of G. For all $n \in \mathbf{N}$, the group $D^n(G)/D^{n+1}(G)$ is a commutative normal (but not in general central) subgroup of $G/D^{n+1}(G)$.

Let $(G_0, G_1, \ldots)$ be a decreasing sequence of subgroups of G such that: (1) $G_0 = G$; (2) for all $i$, $G_{i+1}$ is normal in $G_i$ and $G_i/G_{i+1}$ is commutative. Then $D^i(G) \subset G_i$ for all $i$, as is seen by induction on $i$.