Formally, an amalgam of two groups $G$ and $K$ with core $H$ consists of three groups, $G$, $K$, and $H$, and specified embeddings $i\colon H\to G$ and $j\colon H\to K$. Informally, we think of it as two groups, $G$ and $K$, intersecting a a common subgroup $H$.

   We say that the amalgam is weakly embeddable in a group $M$ if there are embeddings $g\colon G\to M$, $k\colon K\to M$, and $h\colon H\to M$ such that $h=gi$ and $h=kj$. That is, if we can embedd the groups $G$ and $K$ into a group $M$, in such a way that their intersection inside of $M$ contains the subgroup $H$. Note that the definition does not preclude the possibility that some element of $G\setminus H$ be identified with an element of $K\setminus H$ once in $M$.

   We say the amalgam is strongly embeddable if, in addition, we have that $G$ and $K$ intersect only in $H$ inside of $M$.

   By a classic theorem of Schreier (existence of amalgamated free products), if we do not place any extra conditions on $G$, $K$, and $M$, then a strong embedding into some group $M$ is always possible. Also, if $G$ and $K$ are finite, then we can always find a finite group $M$ into which the amalgam is strongly embeddable.

   Suppose, however, that both $G$ and $K$ satisfy additional properties (such as being solvable, or being of exponent $n$, etc.), and we would like that any embedding group $M$ also satisfy these conditions. Can it be done then?

   Then the answer is not nearly so simple. It's been known for a while that in that case the amalgam may not even be weakly embeddable, and so establishing conditions for embeddability becomes and interesting research subject. More on this on the next section.

   Amalgams have played a very important role in many areas of group theory. They are an essential ingredient in HNN-extensions and related studies. They also play a prominent role in combinatorial group theory, and through the Seifert-Van Kampen Theorem, in geometric group theory. Under the name of 'the amalgam method' they are used to study finite simple groups and their classification. They played an important role in the proof of unsolvability of the word problem. And they are used in questions that relate group theory both to Universal Algebra, and to Logic (in the guise of universally closed or existentially closed groups).