Proposition 7

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

(a) $G$ is nilpotent of class $\leq n$. (b) There exists a series of subgroups of $G$:

such that $(G, G^{k'}) \subset G^{k'+1}$ for all $k \in \{1, n\}$.

(c) There exists a subgroup $A$ of $G$ contained in the centre of $G$ such that $G/A$ is nilpotent of class $\leq n - 1$.

(a) $\Rightarrow$ (b): it suffices to take $G^k = C^k(G)$. (b) $\Rightarrow$ (a): by induction on $k$, $C^k(G) \subset G^k$. (a) $\Rightarrow$ (c): it suffices to take $A = C^n(G)$. (c) $\Rightarrow$ (a): let $\pi : G \to G/A$ be the canonical homomorphism; then $\pi(C^n(G)) = C^n(G/A) = \{e\}$ and hence $C^n(G) \subset A$, whence $C^{n+1}(G) = \{e\}$.

More briefly: a group is nilpotent of class $\leq n$ if it can be obtained from the group $\{e\}$ by $n$ successive central extensions.