A group G is said to be capable if it is isomorphic to H/Z(H) for some group H, where Z(H) denotes the center of H. Equivalently, if it is isomorphic to the inner automorphism group of some group.

In his landmark paper on p-groups, Philip Hall remarked that "[t]he question of what conditions a group G must fulfil in order that it may be the central quotient of another group H, G isomorphic to H/Z(H), is an interesting one. But while it is easy to write down a number of necessary conditions, it is not so easy to be sure that they are sufficient."

It was Marshall Hall and J.K. Senior who introduced the term "capable" to describe such groups. In 1938, Baer had made the seminal contribution to the problem, by characterising the finitely generated abelian groups which are capable:

THEOREM (Baer). Let G be a finitely generated abelian group, and write it as a direct sum of cyclic groups of order a_1,...,a_n, with a_i dividing a_{i+1} (a_i=0 for infinite cyclic groups). Then G is capable if and only if n>1 and a_{n-1}=a_{n}.

However, as P. Hall had remarked, the problem is not easy. Some recent progress has been achieved through work of Beyl, Felgner, and Schmid, who characterised capability in terms of the epicenter of a group, and work of Graham Ellis who described the epicenter in terms of the nonabelian tensor square of a group. Recently, M. Bacon and L.C. Kappe have succeeded in characterising the capable 2-generator p-groups of nilpotency class 2, with p an odd prime.

By using the nilpotent product of groups, I have been able to obtain a generalization of Baer's Theorem:

THEOREM Let G be the k-nilpotent product of cyclic p-groups of order p^{a_1},...,p^{a_r}, in non-decreasing order, with p>k. Then G is capable if and only if r>1 and a_{r-1}=a_r.

Also, a necessary condition extending an observation of P. Hall:

THEOREM Let G be a nilpotent p-group of class k, and let x_1,...,x_r be a minimal generating set. Assume that x_i is of order p^{a_i}, with a_1<=...<=a_r. If G is capable, then r>1 and the exponents satisfy a_r<= a_{r-1}+floor((k-1)/(p-1)).

The inequality is best possible. The necessary condition is sufficient for the 2-nilpotent product of cyclic 2-groups.

Using the nilpotent product as a starting point, I have obtained a classification of the capable two-generator p-groups of class two, for any prime p, as well as a number of related results.