Exercise 33, § 6

Let G be a $p$-group and A its automorphism group. (a) Let $(G_i)_{0 \leqslant i \leqslant n}$ be a composition series of G such that $(G_i : G_{i+1}) = p$ for

† In fact this case is impossible, as has been shown by Feit and Thompson, Pac. J. of Math., 13 (1963), 775–1029.

0 \leq i \leq n - 1. Let P be the subgroup of A consisting of the automorphisms u such that, for all i and all x \in G_i, u(x)x^{-1} \in G_{i+1}. Show that, if an element u \in P is of order prime to p, then u = 1 (argue by induction on n). Deduce that P is a p-group.

(b) Conversely, let P be a p-subgroup of A. Show that there exists a composition series $(G_i)_{0 \leq i \leq n}$ of G which is stable under P and such that $(G_i : G_{i+1}) = p$ for $0 \leq i \leq n - 1$; show that the G_i may be chosen to be normal in G.