Group Irregularity Strength of Connected Graphs

We investigate the group irregularity 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. We prove that for any connected graph $G$ of order at least 3, $s_g(G)=n$ if $n\mathrm{\ne }4k+2$ and $s_g(G)\le n+1$ otherwise, except the case of some infinite family of stars.