Exercise 28, § 7

We retain the notation of the preceding exercise. (a) Let $N$ be a subgroup of $\Lambda$. Suppose that, for every $i \in I$, the group $N$ is normal in $G_i$. Prove that $N$ is normal in $G = *_{A}G_i$, and that the canonical homomorphism from $*_{A/N}(G_i/N)$ into $G/N$ is an isomorphism. (b) For every $i \in I$, 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 every $i$. Prove that the canonical homomorphism from $*_{B}H_i$ into $G$ is injective; its image is the subgroup of $G$ generated by the $H_i$.