  We investigate the \textit{group irregularity strength}, $s_g(G)$, of a
graph, i.e. the least integer $k$ such that taking any Abelian group
$\mathcal{G}$ of order $k$, there exists a function $f:E(G)\rightarrow
\mathcal{G}$ so that the sums of edge labels incident with every vertex are
distinct. So far the best upper bound on $s_g(G)$ for a general graph $G$ was
exponential in $n-c$, where $n$ is the order of $G$ and $c$ denotes the number
of its components. In this note we prove that $s_g(G)$ is linear in $n$, namely
not greater than $2n$. In fact, we prove a stronger result, as we additionally
forbid the identity element of a group to be an edge label or the sum of labels
around a vertex.
  We consider also locally irregular labelings where we require only sums of
adjacent vertices to be distinct. For the corresponding graph invariant we
prove the general upper bound: $\Delta(G)+{\rm col}(G)-1$ (where ${\rm col}(G)$
is the coloring number of $G$) in the case when we do not use the identity
element as an edge label, and a slightly worse one if we additionally forbid it
as the sum of labels around a vertex. In the both cases we also provide a sharp
upper bound for trees and a constant upper bound for the family of planar
graphs.

Combinatorics