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Let $p\ne 2$ be a prime and let $G$ be a $p$-group of matrices in ${\mathrm{SL}}_n\left(\mathbb{Z}\right)$, for some integer $n$. In this paper we show that, when $n\<3(p-1)$, a certain subgroup of the cohomology group $H^1(G,{𝔽}_p^n)$ is trivial. We also show that this statement can be false when $n\ge 3(p-1)$. Together with a result of Dvornicich and Zannier (see []), we obtain that any algebraic torus of dimension $n\<3(p-1)$ enjoys a local-global principle on divisibility by $p$.