Exercise 15, § 6

Let $G$ be a group and let $X$ be a subset of $G$ generating $G$. (a) We put $X_1 = X$; for $n \geq 2$, we define $X_n$, by induction on $n$, as the set of commutators $(x, y)$, for $x \in X, y \in X_{n-1}$. Now prove that $X_n = \{e\}$ if and only if G is nilpotent of class $\leq n$. (Reason by induction on $n$. If $X_n = \{e\}$, show that the subgroup H of G generated by $X_{n-1}$ is contained in the center of G, and apply the induction hypothesis to G/H.) (b) Now prove that $C^n(G)$ is the normal subgroup of G generated by $X_n$. Deduce that the image of $X_n$ in $C^n(G)/C^{n+1}(G)$ generates the group $C^n(G)/C^{n+1}(G)$. (c) We take $G = S_m$ ($m \geq 4$) and $X = \{s, t\}$, where s is a transposition and t a cycle of order m. Now prove that $X_2$ does not generate the group $C^2(G) = \mathfrak{A}_m$.