Exercise 36, § 7

Let $(G_1, \ldots, G_n)$ and $(A_1, \ldots, A_{n-1})$ be two sequences of groups; for each $i$, let $\phi_i : A_i \to G_i$ and $\psi_i : A_i \to G_{i+1}$ be two injective homomorphisms; by these homomorphisms $A_i$ can be identified both with a subgroup of $G_i$ and with a subgroup of $G_{i+1}$.

(a) Let $H_1 = G_1, H_2 = H_1 *_{A_1} G_2, \ldots, H_n = H_{n-1} *_{A_{n-1}} G_n$. The group $H_n$ is called the sum of the groups $(G_i)$ amalgamated by the subgroups $(A_i)$; it is denoted by $G_1 *_{A_1} G_2 *_{A_2} \cdots *_{A_{n-1}} G_n$. If $T$ is an arbitrary group, define a bijection between the homomorphisms of $H_n$ into $T$ and the families (t_1, \ldots, t_n) of homomorphisms $t_i : G_i \to T$ such that $t_i \circ \phi_i = t_{i+1} \circ \psi_i$ for $1 \leq i \leq n - 1$.

(b) Each $G_i$ is identified with the corresponding subgroup of $H_n$. Show that $G_i \cap G_{i+1} = A_i$ and that $G_i \cap G_{i+2} = A_i \cap A_{i+1}$ (the latter intersection being taken in $G_{i+1}$).