  Let $W,W'\subseteq G$ be nonempty subsets in an arbitrary group $G$. The set
$W'$ is said to be a complement to $W$ if $WW'=G$ and it is minimal if no
proper subset of $W'$ is a complement to $W$. We show that, if $W$ is finite
then every complement of $W$ has a minimal complement, answering a problem of
Nathanson. This also shows the existence of minimal $r$-nets for every
$r\geqslant 0$ in finitely generated groups. Further, we give necessary and
sufficient conditions for the existence of minimal complements of a certain
class of infinite subsets in finitely generated abelian groups, partially
answering another problem of Nathanson. Finally, we provide infinitely many
examples of infinite subsets of abelian groups of arbitrary finite rank
admitting minimal complements.
