Note on group irregularity strength of disconnected graphs

We investigate the $\mathit{\text{group}}\mathit{\text{irregularity}}\mathit{\text{strength}}$ ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\text{gr}$ of order $s$, there exists a function $f:E(G)\to \text{gr}$ such that the sums of edge labels at every vertex are distinct. So far it was not known if $s_g(G)$ is bounded for disconnected graphs. In the paper we we present some upper bound for all graphs. Moreover we give the exact values and bounds on $s_g(G)$ for disconnected graphs without a star as a component.