  We investigate the \textit{group sum chromatic number} ($\gchi(G)$) of
graphs, i.e. the smallest value $s$ such that taking any Abelian group $\gr$ of
order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums
of edge labels properly colour the vertices. It is known that
$\gchi(G)\in\{\chi(G),\chi(G)+1\}$ for any graph $G$ with no component of order
less than $3$ and we characterize the graphs for which $\gchi(G)=\chi(G)$.
