Exercise 9, § 6

Let $F \to E \to G$ be an extension of finite groups. Suppose no prime number divides both the order of F and the order of G.

(a) Suppose F is solvable. Show that there exists a section $s : G \to E$ and that two such sections are conjugate by an element of F. (Argue by induction on the solvability class of F; where F is commutative, use the preceding Exercise.)

(b) Show the existence of a section† $s : G \to E$ without assuming that F is solvable. (Argue by induction on the order of G. If $p$ divides the order of F, choose a Sylow $p$-subgroup P of F and consider its normalizer N in E. The image of N in G is equal to G, cf. Exercise 25. If $N \neq E$, conclude by means of the induction hypothesis; if $N = E$, use (a) and the induction hypothesis applied to $F/P \to E/P \to G$.)