I have some basic questions about properties of free product amalgamation of groups. Both can be phrased inside some fixed group $G$, which I am thinking of as having some very large infinite cardinality (but I don't think this is important).
Given subgroups $A,B\leq G$, I will use $\langle AB\rangle$ to denote the subgroup of $G$ generated by $A\cup B$ (equivalently, by the set product $AB$).
Given subgroups $A,B,C\leq G$, with $C\leq A\cap B$, I will use $A\ast_C B$ to denote the usual free product amalgamation of $A$ and $B$ over $C$ (as in this link).
Edit (re: Derek Holt's very good suggestion) Given subgroups $A,B,C\leq G$, I will write $\langle ABC\rangle\cong\langle AC\rangle\ast_C\langle BC\rangle$ to mean that the identity map on $A\cup B\cup C$ generates an isomorphism between the two groups.
Question #1
Fix subgroups $D\leq C\leq B\leq G$ and $A\leq G$. Consider the following statements.
1) $\langle AB\rangle \cong \langle AC\rangle \ast_C B$ and $\langle AC\rangle\cong\langle AD\rangle\ast_D C$
2) $\langle AB\rangle \cong \langle AD\rangle\ast_D B$
What is the relationship between (1) and (2)? Are they equivalent?
Question #2
Fix subgroups $A,A',B,C\leq G$, with $C\leq A\cap A'\cap B$. Assume:
- $\langle AB\rangle\cong A\ast_C B$,
- $\langle A'B\rangle\cong A'\ast_C B$, and
- there is an isomorphism $\phi:A\longrightarrow A'$ such that $\phi(c)=c$ for all $c\in C$.
Define $\psi:\langle AB\rangle\longrightarrow\langle A'B\rangle$ such that $\psi$ extends $\phi$ and $\psi(b)=b$ for all $b\in B$. Is $\psi$ a well-defined isomorphism?
I imagine that these problems are not difficult for someone used to this kind of thing; so helpful answers would also include references to sufficiently similar exercises in standard texts.