Proposition 5

Every isomorphism of the topological group $\mathbf{T}^n$ onto one of its subgroups is an automorphism of $\mathbf{T}^n$ which is obtained by passing to the quotient from a linear mapping $u$ of $\mathbf{R}^n$ onto itself which, restricted to $\mathbf{Z}^n$, is an automorphism of $\mathbf{Z}^n$.

Equivalently (§ 1, no. 1), if $u(e_i) = \sum_{j=1}^n a_{ij} e_j$, the $a_{ij}$ must be integers such that $\det(a_{ij}) = \pm 1$. In particular, for $n = 1$: