Exercise 18, § 4

The upper central series ($Z_iG$) of the group $G$ is defined inductively as follows: (i) $Z_iG = \{e\}$ if $i \leqslant 0$; (ii) $Z_iG/Z_{i-1}G$ is the centre of $G/Z_{i-1}G$. Then $\{e\} = Z_0G \subset Z_1G \subset \cdots$ and $Z_1G$ is the centre of $G$; the $Z_iG$ are characteristic subgroups of $G$. (a) Show that $G$ is nilpotent of class $\leq c$ if and only if $G = Z_cG$. (b) Show that $(C^nG, Z_mG) \subset Z_{m-n}G$. (c) Suppose that $G$ is nilpotent and $P$-torsion-free (where $P$ is a set of prime numbers, cf. Exercise 14). Show that the $Z_iG$ are $P$-saturated. (It suffices to verify this for $Z_1G$; if $n$ is a $P$-integer and $g \in G$ is such that $g^n \in Z_1G$, then $xgx^{-1} = g^n$ for all $x \in G$, whence $xgx^{-1} = g$ by Exercise 14 (c) and it follows that $g$ belongs to $Z_1G$.)