Exercise 20, § 4

(a) Let G be a group, the direct product of two subgroups A and B. Let $C_A$ be the centre of A and let N be a normal subgroup of G such that $N \cap A = \{1\}$. Show that $N \subset C_A \times B$.

(b) Let $(S_i)_{i \in I}$ be a finite family of non-commutative simple groups. Show that every normal subgroup of $\prod_{i \in I} S_i$ is equal to one of the partial products $\prod_{i \in J} S_i$ where J is a subset of I.