Exercise 32, § 6

Let $p$ be a prime number and G a $p$-group. Let $G^* = G^p D(G)$ be the subgroup of G generated by D(G) and the $x^p, x \in G$. (a) Show that $G^*$ is the intersection of the kernels of the homomorphisms of G into $\mathbf{Z}/p\mathbf{Z}$. (b) Show that a subset S of G generates G if and only if the image of S in $G/G^*$ generates $G/G^*$. Deduce that, if $(G:G^*) = p^n$, the integer $n$ is the minimum number of elements in a subset of G which generates G; in particular, G is cyclic if and only if $n \leqslant 1$. (c) Let $u$ be an automorphism of G of order prime to $p$ and let $\bar{u}$ be the corresponding automorphism of $G/G^*$. Show that $\bar{u} = 1$ implies $u = 1$.