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

Realization of digraphs in Abelian groups and its consequences