What is Amalgamation My question is what exactly is free product of groups with amalgamation? 
I came across this term in algebraic topology.
Is it possible to explain it at a level that an undergraduate with knowledge of basic group theory  can understand?
Some examples would be fine too.
Thanks! 
 A: It is better to illustrate first through examples: let $G_1=\langle x\rangle \cong \mathbb{Z}_4$ and $G_2=\langle y\rangle\cong \mathbb{Z}_6$. We see: 


*

*$\langle x^2\rangle$ is a subgroup of order $2$ in $G_1$; 

*$\langle y^3\rangle$ is a subgroup of order $2$ in $G_2$.
Both are isomorphic subgroups; but not "same" (coincide). We try to put the groups $G_1$ and $G_2$ in a bigger group where $\langle x^2\rangle$ and $\langle y^3\rangle$ will become "same"; and we do "that only". 
Consider words in $x$ and $y$ with "only" condition that $x^4=1$ and $y^6=1$. This is nothing but "free" product of $\langle x\rangle$ and $\langle y\rangle$. Next, we pose "one more (and only one) condition" to identify groups $\langle x^2\rangle$ and $\langle y^3\rangle$. A simple condition is $x^2=y^3$; i.e. whenever a word in $x,y$ contains a term $x^2$ we can write there $y^3$. Thus, we have a new group:
$$\langle x,y\colon x^4, y^6, x^2=y^3\rangle.$$
In this group, the word $x^3y^3$ can be written as 
$$x^3y^3=x.x^2.y^3=x.y^3.y^3=xy^6=x$$
simpler expression, because of gluing $x^2$ and $y^3$.
In other words, we have "glued" (amal-gum-ated) groups $\langle x\rangle$ and $\langle y\rangle$ by cyclic group of order $2$ in both.
Now instead of cyclic groups of order $4$ and $6$, if we take cyclic groups of order say $3$ and $8$, then these orders are relatively prime, the only subgroup common in them is identity. But when we put these groups in a larger group, as the identity element in larger group is unique, we will not have to do "gluing" of identity elements in $\mathbb{Z}_3$ and $\mathbb{Z}_8$. In this case, we get simply "free" product of two cyclic groups of order $3$ and $8$; it has presentation:
$$\langle a,b\colon a^3, b^8\rangle.$$
With such simple examples, the "abstract" theory of amalgamation will become clear easily.
