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Given $S \in GL_n(\mathbb{R}^n)$.

Show that $x \mapsto Sx$ is an orientation preserving diffeomorphism on $\mathbb{T}^n$ if and only if $S \in SL_n(\mathbb{Z}^n)$.

I'm working on the only if part. Orientation perserving implies $\det(S) > 0$.

How can I prove that we must have $S \in SL_n(\mathbb{Z}^n)$. I can see that when $S \in SL_n(\mathbb{R}^n)$ is not enough to be surjective, but can't prove that $S \in SL_n(\mathbb{Z}^n)$.

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As we showed here the simple fact that $S$ induces a diffeomorphism on $\mathbb T^n$ forces $S \in \text{GL}_n(\mathbb Z)$. Then $\text{det}(S) \in \{\pm 1\}$, hence $\text{det}(S) = 1$.