Central Automorphisms and Inner Automorphisms in Finitely Generated Groups

Let G be a group and Aut(c)(G) be the group of all central automorphisms of G. We know that in a finite p-group G, Aut(c)(G)=Inn(G) if and only if Z(G)=G and Z(G) is cyclic. But we shown that we cannot extend this result for infinite groups. In fact, there exist finitely generated nilpotent groups of class 2 in which G =Z(G) is infinite cyclic and Inn(G)<C*=Aut(c)(G). In this article, we characterize all finitely generated groups G for which the equality Aut(c)(G)=Inn(G) holds.