We will prove that, for any abelian group $G$, the canonical (surjective and continuous) mapping $\boldsymbol{ \beta}G \to { \frak b}G$ from the Stone-\v{ C}ech compactification $\boldsymbol{ \beta}G$ of $G$ to its Bohr compactfication ${ \frak b}G$ is a homomorphism with respect to the semigroup operation on $\boldsymbol{ \beta}G$, extending the multiplication on $G$, and the group operation on ${ \frak b}G$. Moreover, the Bohr compactification ${ \frak b}G$ is canonically isomorphic (both in algebraic and topological sense) to the quotient of $\boldsymbol{ \beta}G$ with respect to the least closed congruence relation on $\boldsymbol{ \beta}G$ merging all the Schur ultrafilters on $G$ into the unit of $G$.
Categories: math.GN