Derived subgroups and centers of capable groups

Abstract:

A group $G$ is said to be capable if it is isomorphic to the central factor group $H/\mathbf {Z}(H)$ for some group $H$. It is shown in this paper that if $G$ is finite and capable, then the index of the center $\mathbf {Z}(G)$ in $G$ is bounded above by some function of the order of the derived subgroup $G’$. If $G’$ is cyclic and its elements of order $4$ are central, then, in fact, $|G:\mathbf {Z}(G)| \le |G’|^{2}$.