  We investigate the \textit{group irregular strength} $(s_g(G))$ of graphs,
i.e the smallest value of $s$ such that for any Abelian group $\Gamma$ of order
$s$ exists a function $g\colon E(G) \rightarrow \Gamma$ such that sums of edge
labels at every vertex is distinct. We give results for bound and exact values
of $(s_g(G))$ for graphs without small stars as components.
