  Can we do a topological study of various classes of normal subgroups endowed
with a hull-kernel-type topology? In this paper, we have provided an answer to
this question. We have introduced as well a new class of normal subgroups
called primitive subgroups. Separation axioms, compactness, connectedness, and
continuities of these spaces have been studied. We have concluded with the
question of determining spectral spaces among them.
