Group irregularity strength of connected graphs

We investigate the group irregularity strength ( $s_g(G)$ ) of graphs, that is, we find the minimum value of $s$ such that for any Abelian group $\mathcal{G}$ of order $s$ , there exists a function $f:E(G)\to \mathcal{G}$ such that the sums of edge labels at every vertex are distinct. We prove that for any connected graph $G$ of order at least $3$ , $s_g(G)=n$ if $n\ne 4k+2$ and $s_g(G)\le n+1$ otherwise, except the case of an infinite family of stars. We also prove that the presented labelling algorithm is linear with respect to the order of the graph.