  A global offensive alliance in a graph $G$ is a set $S$ of vertices with the
property that every vertex not belonging to $S$ has at least one more neighbor
in $S$ than it has outside of $S$. The global offensive alliance number of $G$,
$\gamma_o(G)$, is the minimum cardinality of a global offensive alliance in
$G$. A set $S$ of vertices of a graph $G$ is a dominating set for $G$ if every
vertex not belonging to $S$ has at least one neighbor in $S$. The domination
number of $G$, $\gamma(G)$, is the minimum cardinality of a dominating set of
$G$. In this work we obtain closed formulas for the global offensive alliance
number of several families of Cartesian product graphs, we also prove that
$\gamma_o(G\square H)\ge \frac{\gamma(G)\gamma_o(H)}{2}$ for any graphs $G$ and
$H$ and we show that if $G$ has an efficient dominating set, then
$\gamma_o(G\square H)\ge \gamma(G)\gamma_o(H).$ Moreover, we present a
Vizing-like conjecture for the global offensive alliance number and we prove it
for several families of graphs.
