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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 $H$, $G\cong H/Z(H)$, 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 $a_1,\ldots,a_n$, with $a_i|a_{i+1}$ ($a_i=0$ for infinite cyclic groups). Then $G$ is capable if and only if $n\gt 1$ 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 $p^{a_1},\ldots,p^{a_r}$, in non-decreasing order, with $p\gt k$. Then $G$ is capable if and only if $r\gt 1$ 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 $x_1,\ldots,x_r$ be a minimal generating set. Assume that $x_i$ is of order $p^{a_i}$, with $a_1\leq \cdots \leq a_r$. If $G$ is capable, then $r\gt 1$ and the exponents satisfy $a_{r}\leq a_{r-1}+\lfloor \frac{k-1}{p-1}\rfloor$.

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.

More recently, with coauthors Azhana Ahmad and Robert F. Morse, we completely re-did the classification of two-generator $p$-groups of class two (which turned out to be incomplete for all primes, and an incorrect claim of disjointness of families for $p=2$); and used the classification to verify the description of the capable groups in the class. We also computed a number of homological functors such as the tensor and exterior squares, and the Schur multiplier, and obtained a formula to count the groups in this class. The work appears in Publ. Math Debrecen and in the Contemporary Mathematics volume entitled Computational Group Theory and the Theory of Groups II.


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