Given groups $G\_1, G\_2, G\_3$ and injections $A\_1 \to G\_1$ and $A\_1 
\to G\_2$ , from $A\_2 \to G\_2$ and $A\_2 \to G\_3$, let $G\_1 \*\_{A\_1} \*G\_2 \*\_{A\_2} G\_3$ be the amalgam formed these groups and maps. 
Then is it true that $G\_1 \*\_{A\_1} \*G\_2 \*\_{A\_2} G\_3$ is the same as (G\_1 \*\_{A\_1} G\_2 ) \*\_{A\_2} G\_3. If yes, how do we see this?



is amalgamation of groups associative
Amalgamation of groups is a categorical construction known as a "pushout": <http://en.wikipedia.org/wiki/Pushout_(category_theory)>


By general category theory, pushouts are associative up to unique isomorphism, i.e., the two things you wrote are isomorphic in a unique way (subject to commuting with the inclusions from A\_i, etc.)


