  Tkachenko and Yaschenko [34] characterized the abelian groups G such that all
proper unconditionally closed subsets of G are finite, these are precisely the
abelian groups G having cofinite Zariski topology (they proved that such a G is
either almost torsion-free or of prime exponent). The authors connected this
fact to Markov's notion of potential density and the existence of pairs of
independent group topologies. Inspired by their work, we examine the class C of
groups having cofinite Zariski topology in the general case, obtaining a number
of very strong restrictions on these groups in the non-abelian case which
suggest the bold conjecture that a group with cofinite Zariski topology is
necessarily either abelian or finite. We show that Tkachenko-Yaschenko theorem
fails in the non-abelian case and we offer a natural counterpart in the general
case using a partial Zariski topology and an appropriate stronger version of
the property almost torsion-free.
