  A precompact group topology $\tau$ on an abelian group $G$ is called {\em
single sequence characterized} (for short, {\em ss-characterized}) if there is
a sequence $\mathbf{u}= (u_n)$ in $G$ such that $\tau$ is the finest precompact
group topology on $G$ making $\mathbf{u}=(u_n)$ converge to zero. It is proved
that a metrizable precompact abelian group $(G,\tau)$ is $ss$-characterized iff
it is countable. For every metrizable precompact group topology $\tau$ on a
countably infinite abelian group $G$ there exists a group topology $\eta$ such
that $\eta$ is strictly finer than $\tau$ and the groups $(G,\tau)$ and
$(G,\eta)$ have the equal Pontryagin dual groups. We give a complete
description of all $ss$-characterized precompact abelian groups modulo
countable $ss$-characterized groups from which we derive:
  (1) No infinite pseudocompact abelian group is $ss$-characterized.
  (2) An $ss$-characterized precompact abelian group is hereditarily
disconnected.
