  We investigate the \textit{group irregularity strength} ($s_g(G)$) of graphs,
i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of
order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums
of edge labels at every vertex are distinct. We give the exact values and
bounds on $s_g(G)$ for chosen families of disconnected graphs. In addition we
present some results for the \textit{modular edge gracefulness} $k(G)$, i.e.
the smallest value of $s$ such that there exists a function $f:E(G)\rightarrow
\zet_s$ such that the sums of edge labels at every vertex are distinct.

Combinatorics