Algebraic characterizations of generating and affinely generating $Γ$-magic maps and $Γ$-distance magic labelings on regular graphs

For an abelian group $Γ$ of order $n$, a graph $G$ of order $n$ is $Γ$-distance magic if it admits a bijection $V(G)\toΓ$ whose open-neighborhood sums are constant, and group distance magic if this holds for every such $Γ$. More generally, for any finite abelian $Γ$, a $Γ$-magic map is a map $f\colo