Let G=(V,E) be a graph. For a non-empty subset of vertices S⊆V,
and vertex v∈V, let δS(v)=∣{u∈S:uv∈E}∣ denote the
cardinality of the set of neighbors of v in S, and let Sˉ=V−S.
Consider the following condition: {equation}\label{alliancecondition}
\delta_S(v)\ge \delta_{\bar{S}}(v)+k, \{equation} which states that a vertex
v has at least k more neighbors in S than it has in Sˉ. A set
S⊆V that satisfies Condition (\ref{alliancecondition}) for every
vertex v∈S is called a \emph{defensive} k-\emph{alliance}; for every
vertex v in the neighborhood of S is called an \emph{offensive}
k-\emph{alliance}. A subset of vertices S⊆V, is a \emph{powerful}
k-\emph{alliance} if it is both a defensive k-alliance and an offensive (k+2)-alliance. Moreover, a subset X⊂V is a defensive (an offensive or
a powerful) k-alliance free set if X does not contain any defensive
(offensive or powerful, respectively) k-alliance. In this article we study
the relationships between defensive (offensive, powerful) k-alliance free
sets in Cartesian product graphs and defensive (offensive, powerful)
k-alliance free sets in the factor graphs