Realization of digraphs in Abelian groups and its consequences

Let $overrightarrow{G}$ be a directed graph with no component of orderless than~$3$, and let $Gamma$ be a finite Abelian group such that $|Gamma|geq 4|V(overrightarrow{G})|$ or if $|V(overrightarrow{G})|$ is large enough with respect to an arbitrarily fixed $varepsilon>0$ then $|Gamma|geq (1+varepsilon)|V(overrightarrow{G})|$. We show that there exists an injective mapping $varphi$ from $V(overrightarrow{G})$ to the group $Gamma$ such that $sum_{xin V(C)}varphi(x) = 0$ for every connected component $C$ of $overrightarrow{G}$, where $0$ is the identity element of $Gamma$. Moreover we show some applications of this result to group distance magic labelings.