Say we have an amalgam of groups G and K over core H, and that G and K belong to some specified class of groups, such as all finite groups, solvable groups, simple groups, etc. It makes sense to ask whether we can find a group M into which to embedd the amalgam (at least weakly), with M also belonging to that same class.

   For the class of abelian groups, or even of abelian groups of a given exponent, the result is easy and was well known. If the class is all finite groups, then Schreier also proved that an amalgam of finite groups can always be strongly embedded into a finite group. For other classes, though the situation is not nearly so good.

   Thus, for example, not every amalgam of solvable groups can be embedded (even weakly) into a solvable group. Not every amalgam of finite p-groups can be embedded into a finite p group, etc. In these classes, what we want is to give conditions which are both necessary and sufficient to guarantee that the amalgam is weakly and/or strongly embeddable in that class.

   The types of conditions that have traditionally been obtained fall roughly into four classes, which I now describe:

   1. CIM Conditions. "CIM" stands for "commutator, intersection and multiplication." The idea is to take the groups G, K, and H, together with the commutator subgroups of them, and to form a sort of incomplete algebra of subgroups by using the operations of taking intersection of subgroups, multiplying subgroups, or taking the commutator of subgroups. Then, the condition is to be expressed via equalities of these subgroups. The archetypical example of a CIM condition for embeddability is G. Higman's theorem on strong embeddability of amalgams of finite p-groups into a finite p-group. (See Amalgams of p-groups, Graham Higman, J. Algebra 1 (1964), 301-305).

   Unfortunately, CIM conditions have severe limitations. For example, it is known that they cannot suffice to establish embeddability of even amalgams of nilpotent groups of class two into a solvable group. (See On certain embeddability criteria for group amalgams, B.H. Neumann and James Wiegold, Pub. Math. Debrecen 9 (1962), 57-64.) This leads us to the second kind of conditions

   2. Internal conditions. These are conditions which depend only on the amalgam itself, with no invocation of other groups or morphisms. They are usually in the form of combinations of CIM-type conditions (although extended to include inclusions among the groups, not just equalities), and equations that have to be satisfied. Typical examples are Berthold Maier's conditions for weak and for strong embeddability of amalgams of nilpotent groups of class two. See Amalgame nilpotenter Gruppen der Klasse zwei B. Maier, Publ. Math. Debrecen 31 (1985), 57-70; and Amalgame nilpotenter Grupper der Klasse zwei II Publ. Math. Debrecen 33 (1986), 43-52. I have generalized both of them for arbitrary subvarieties of nilpotent groups of class two. You can see the result in my preprint Amalgams of nilpotent groups of class two.

   3. External conditions. These are conditions that do invoke other groups or morphisms. A typical example is Felix Leinen's theorem on embeddability of an amalgam of solvable groups into solvable groups. See An amalgamation theorem for solvable groups, Felix Leinen, Canadian Math. Bull. 30 (1987) no. 1, 9-18.

   External conditions, although important theoretically, tend to be very hard to work with, even when trying to apply them to specific groups.

   The final broad type of condition is:

   4. Conditions on the core. These are conditions which are given exclusively on the group H to imply embeddability. A very silly example would be "H is trivial." Although they will not characterize embeddability in a class, we can hope to characterize the amalgamation bases. More on amalgamation bases in the next section.

   If every amalgam of V groups is weakly embeddable in V we say that V has the weak amalgamation property. Similarly, we say V has the strong amalgamation property if every amalgam of V groups is strongly embeddable into a V group. Clearly, the strong amalgamation property implies the weak amalgamation property. The gap between them is filled by the special amalgams.

   An amalgam is special if G and K are isomorphic over H; that is, there is an isomorphism between G and K that identifies the two copies of H. In this case, the amalgam is always weakly embeddable, since we may simply map G to K via the given isomorphism, and K to itself via the identity map. In this case, every element of the groups gets identified (this is as bad as a weak embedding can get). Because of this, we are usually interested in embedding a special amalgam "as strongly as possible", i.e. to find the smallest subgroup of G containing H over which the amalgam is strongly embeddable.

   If every special amalgam in V is strongly embeddable, we say that V has the special amalgamation property. It turns out that this property "measures" the gap between the weak and strong amalgamation property, in the sense that the following result holds in general:

   Theorem. A class V has the strong amalgamation property if and only if it has both the weak and special amalgamation properties.

   Because of this, studying special amalgams provides important information and insight into amalgams in a given class which does not have the strong amalgamation property, or where not every weakly embeddable amalgam is strongly embeddable.