# On minimal complements in groups

## TL;DR

This paper investigates the existence and properties of minimal complements in groups, proving their existence for finite subsets and providing conditions and examples for infinite subsets, thus advancing the understanding of group coverings.

## Contribution

It proves every complement of a finite subset has a minimal complement in any group, and characterizes minimal complements for certain infinite subsets in finitely generated abelian groups.

## Key findings

- Every complement of a finite subset has a minimal complement.
- Existence of minimal r-nets in finitely generated groups for all r ≥ 0.
- Infinite subsets of abelian groups can admit minimal complements under certain conditions.

## Abstract

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.