Summary: We say that a finite abelian group \(\Gamma\) has the constant-sum-partition property into \(t\) sets (CSP\((t)\)-property) if for every partition \(n=r_1+r_2+\dots +r_t\) of \(n\), with \(r_i\geq 2\) for \(2\leq i\leq t\), there is a partition of \(\Gamma\) into pairwise disjoint subsets \(A_1,A_2,\dots ,A_t\), such that \(A_i=r_i\) and for some \(\nu \in \Gamma\), \(\sum_{a\in A_i}a=\nu\) for \(1\leq i\leq t\). For \(\nu=g_0\) (where \(g_0\) is the identity element of \(\Gamma\)) we say that \(\Gamma\) has zero-sum-partition property into \(t\) sets (ZSP(t)-property). A \(\Gamma\)-distance magic labeling of a graph \(G=(V,E)\) with \(V=n\) is a bijection from \(V\) to an abelian group \(\Gamma\) of order \(n\) such that the weight \(w(x)=\sum_{y\in N(x)}\ell(y)\) of every vertex \(x\in V\) is equal to the same element \(\mu \in \Gamma\), called the magic constant. A graph \(G\) is called a group distance magic graph if there exists a \(\Gamma\)-distance magic labeling for every abelian group \(\Gamma\) of order \(V(G)\). In this paper we study the CSP\((3)\)-property of \(\Gamma\), and apply the results to the study of group distance magic complete tripartite graphs.