Example

(1) A group is solvable of class 0 (resp. $\leq 1$) if and only if it is reduced to $\{e\}$ (resp. is commutative).

(2) Every nilpotent group of class $\leq 2^n - 1$ is solvable of class $\leq n$; this follows from the relation $D^n(G) \subset C^{2^n}(G)$ proved above.

(3) Let G be a solvable group of class $\leq n$. Every subgroup (resp. quotient group) of G is solvable of class $\leq n$ (proof analogous to that of no. 3, Example 3).

(4) If G is a solvable group of class $p$ and F is a solvable group of class $q$, every extension E of G by F is a solvable group of class $\leq p + q$. For, let $\pi : E \to G$ be the projection; then $\pi(D^p(E)) \subset D^p(G) = \{e\}$ and therefore $D^p(E) \subset F$; it follows that $D^{p+q}(E) = D^q(D^p(E)) \subset D^q(F) = \{e\}$.

(5) The symmetric group $S_n$ is solvable if and only if $n < 5$ (cf. § 5, Exercises 10 and 16).

(6) If A is a commutative ring, the upper triangular group T(n, A) is solvable but not in general nilpotent.