Let \(W,W'\subseteq G\) be non-empty subsets in an arbitrary group G. The set \(W'\) is said to be a complement to W if \(W\cdot W'=G\) 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.

令\（W，W'\ subseteq G \）是任意组G中的非空子集。该组\（W“\）被说成是一种补体W¯¯如果\（W \ CDOT W” = G \），它是最小的，如果没有适当的子集\（W'\）是补体w ^。我们发现，如果w ^是有限的，然后每个补W¯¯具有最小的补充，回答内桑森的问题。这也表明每个\（r \ geqslant 0 \）都存在最小r -nets在有限生成的组中。此外，我们给出了在有限生成的阿贝尔群中某类无限子集的最小补的存在的充要条件，部分回答了纳森森的另一个问题。最后，我们提供了无限个例子，其中包含最小补码的任意有限秩的阿贝尔群的无限子集。