  We investigate the \textit{edge group irregularity strength} ($es_g(G)$) of
graphs, i.e. the smallest value of $s$ such that taking any Abelian group
$\mathcal{G}$ of order $s$, there exists a function $f:V(G)\rightarrow
\mathcal{G}$ such that the sums of vertex labels at every edge are distinct. In
this note we provide some upper bounds on $es_g(G)$ as well as for edge
irregularity strength $es(G)$ and harmonious order $\rm{har}(G)$.
