Definition 5

Let G be a group. The lower central series of G is the sequence $(C^n(G))_{n \geq 1}$ of subgroups of G defined inductively by:

Let $f : G \to G'$ be a group homomorphism. It is seen, by induction on $n$, that $f(C^n(G)) \subset C^n(G')$ and that, if $f$ is surjective, $f(C^n(G)) = C^n(G')$. In particular, for all $n \geq 1$, $C^2(G)$ is a characteristic (and hence normal) subgroup of $G$. For all $n \geq 1$, $C^n(G)/C^{n+1}(G)$ is contained in the centre of $G/C^{n+1}(G)$.

Let $(G_1, G_2, \ldots)$ be a decreasing sequence of normal subgroups of $G$ such that (1) $G_1 = G$; (2) for all $i$, $G_i/G_{i+1}$ is contained in the centre of $G/G_{i+1}$. Then $C^i(G) \subset G_i$, as is seen by induction on $i$.

Now

For, if this relation is denoted by $(F_{m,n})$, it follows from $(F_{m,n})$, by no. 2, Proposition 5, that Hence $((F_{m,n}) \text{ and } (F_{m+1,n})) \Rightarrow (F_{m,n+1})$. As $(F_{m,1})$ and $(F_{1,n})$ are obvious $(F_{m,n})$ follows by induction.