Exercise 9, § 4

Let $G$ be a group which does not consist only of the identity element. Suppose that there exists a finite subset $S$ of $G$ such that $G$ is generated by $S$ (resp. by the elements $gsg^{-1}, g \in G, s \in S$). Let $\mathfrak{N}$ be the set of subgroups of $G$ (resp. of normal subgroups of $G$) distinct from $G$. Show that $\mathfrak{N}$, ordered by inclusion, is inductive. Deduce the existence of a normal subgroup $H$ of $G$ such that $G/H$ is simple.