Let $\alpha: C \rightarrow A$ and $\beta: C \rightarrow B$ be monomorphisms and let $G$ be the corresponding amalgamated product of $A,B$ over $C$. Let $A_1$ be a subgroup of index $2$ of $A$, such that $\alpha(C) \subset A_1$ but $\alpha (C) \neq A_1$. Show that the subgroup $H$ of $G$ generated by $A_1$ and $B$ is an amalgamated product over $C$. What is the index of $H$ in $G$.
I am looking for a detailed solution to this question. To me, it looks like showing that $H$ is an amalgamated product over $C$ is trivial, and this is making me think I am severely misunderstanding something here: The presentation for $G$ includes relations $\{\alpha(c) = \beta(c)|\, \forall c \in C \}$ and since $\alpha(C) \subset A_1$ and $H$ is viewed as a subgroup of $G$, then all those relations are already present, so the presentation for $H$ would indeed be that of such of an amalgamated product... For finding the index of $H$ in $G$, it thought that i could define a homomorphism from $G$ to $\mathbb{Z}_2$ by sending $b \in B$ to $0$ and $a\in A$ to $aA_1$. Unfortunately this is may not be a homomorphism due to the relations in $A$. I am unsure about how to proceed. If you cannot provide a detailed solution, I would also appreciate some thoughts on how I could approach 'Proving that something is an amalgamated product' and 'Thinking about the index of a subgroup' For example, in the second case, i like finding homomorphisms whose kernels are the subgroup i'm interested in.