  Many properties of groups can be defined by the existence of a particular
normal series. The classic examples being solvability, supersolvability and
nilpotence. Among the nilpotent groups are the so-called nested GVZ-groups ---
groups where the centers of the irreducible characters form a chain, and where
every irreducible character vanishes off of its center. In this paper, we show
that nested GVZ-groups can be characterized by the existence of a certain
ascending central series, or by the existence of a certain descending central
series.
