Exercise 33, § 6

Let G be a $p$-group and let A be the group of its automorphisms. a) Let $(G_i)_{0 \leqslant i \leqslant n}$ be a composition sequence of G such that $(G_i : G_{i+1}) = p$ for $0 \leqslant i \leqslant n-1$. Let P be the subgroup of A formed by the automorphisms $u$ such that, for all $i$ and all $x \in G_i$, one has $u(x)x^{-1} \in G_{i+1}$. Now prove that, if an element $u \in P$ is of order prime to $p$, one has $u = 1$ (reason by induction on $n$). Deduce that P is a $p$-group. b) Conversely, let P be a $p$-subgroup of A. Now prove that there exists a composition sequence $(G_i)_{0 \leqslant i \leqslant n}$ of G, stable under P, such that $(G_i : G_{i+1}) = p$ for $0 \leqslant i \leqslant n-1$; prove that one can choose the $G_i$ distinguished in G.