Exercise 18, § 4

(a) Let G be a group with operators generated by the union of a family $(H_i)_{i \in I}$ of simple normal stable subgroups of G. Show that there exists a subset J of I such that G is the restricted sum of the family $(H_i)_{i \in J}$ (apply Zorn’s Lemma to the set of subsets K of I such that the subgroup generated by the union of the $H_i$ for $i \in K$ is the restricted sum of this family).

(b) Let A be a normal stable subgroup of G. Show that there exists a subset $J_A$ of I such that G is the restricted sum of A and the $H_i$ for $i \in J_A$. Deduce that A is isomorphic to the restricted sum of a subfamily of the $H_i$.