Realization of digraphs in Abelian groups and its consequences

Let G -> be a directed graph of order n with no component of order less than 3, and let Gamma be a finite Abelian group such that divide Gamma divide >= 2n+2n or if n is large enough with respect to an arbitrarily fixed epsilon>0 then divide Gamma divide >=(1+epsilon)n. We show that there exists an injective mapping phi from V(G ->) to the group Gamma such that n-ary sumation x is an element of V(C ->)phi(x)=0 for every connected component C -> of G ->, where 0 is the identity element of Gamma. Moreover we show some applications of this result to group distance magic labelings.