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 $Γ$ of order $s$ exists a function $g\colon E(G) \rightarrow Γ$ 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.
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