Exercise 28, § 7

We preserve the notation of the preceding exercise.

(a) Let $N$ be a subgroup of $A$. Suppose that, for all $i \in I$, the group $N$ is normal in $G_i$. Show that $N$ is normal in $G = *_A G_i$ and that the canonical homomorphism of $*_{A/N} G_i / N$ into $G / N$ is an isomorphism.

(b) For all $i \in I$, let $H_i$ be a subgroup of $G_i$ and let $B$ be a subgroup of $A$. Suppose that $H_i \cap A = B$ for all $i$. Show that the canonical homomorphism of $*_B H_i$ into $G$ is injective; its image is the subgroup of $G$ generated by the $H_i$.