Exercise 15, § 6

Let G be a group and X a subset of G generating G. (a) Write $X_1 = X$; for $n \geq 2$, define $X_n$, by induction on n, as the set of commutators $(x, y), x \in X, y \in X_{n-1}$. Show that $X_n = \{e\}$ if and only if G is nilpotent of class $\leq n$. (Argue by induction on n. If $X_n = \{e\}$, show that the subgroup H of G generated by $X_{n-1}$ is contained in the centre of G and apply the induction hypothesis to G/H.) (b) Show that $C^n(G)$ is a 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) Take $G = S_m$ ($m \geq 4$) and $X = \{s, t\}$, where s is a transposition and t a cycle of order m. Show that $X_2$ does not generate the group $C^2(G) = \mathfrak{A}_m$.