Extensions of the Moser-Scherck-Kemperman-Wehn Theorem

TL;DR

This paper extends the Moser-Scherck-Kemperman-Wehn Theorem to cofinite subsets in reflexive relations with transitive automorphism groups, providing new bounds and applications in group theory.

Contribution

It generalizes the classical theorem to cofinite sets and explores implications in group automorphisms and relations.

Findings

Extended the theorem to cofinite subsets.

Established bounds for reflexive relations with transitive automorphisms.

Provided applications in group theory and relations.

Abstract

Let $\Gamma =(V,E)$ be a reflexive relation having a transitive group of automorphisms and let $v\in V.$ Let $F$ be a subset of $V$ with $F\cap \Gamma ^-(v)=\{v\}$. (i) If $F$ is finite, then $| \Gamma (F)\setminus F|\ge |\Gamma (v)|-1.$ (ii) If $F$ is cofinite, then $| \Gamma (F)\setminus F|\ge |\Gamma ^- (v)|-1.$ In particular, let $G$ be group, $B$ be a finite subset of $G$ and let $F$ be a finite or a cofinite subset of $G$ such that $F\cap B^{-1}=\{1\}$. Then $| (FB)\setminus F|\ge |B|-1.$ The last result (for $F$ finite), is famous Moser-Scherck-Kemperman-Wehn Theorem. Its extension to cofinite subsets seems new. We give also few applications.