  The notion of minimal complements was introduced by Nathanson in 2011. Since
then, the existence or the inexistence of minimal complements of sets have been
extensively studied. Recently, the study of inverse problems, i.e., which sets
can or cannot occur as minimal complements has gained traction. For example,
the works of Kwon, Alon--Kravitz--Larson, Burcroff--Luntzlara and also that of
the authors, shed light on some of the questions in this direction. These works
have focussed mainly on the group of integers, or on abelian groups. In this
work, our motivation is two-fold: (i) to show some new results on the inverse
problem, (ii) to concentrate on the inverse problem in not necessarily abelian
groups. As a by-product, we obtain new results on non-minimal complements in
the group of integers and more generally, in any finitely generated abelian
group of positive rank and in any free abelian group of positive rank.
Moreover, we show the existence of uncountably many subsets in such groups
which are "robust" non-minimal complements.

Combinatorics;Group Theory