Free actions of finite groups on the 3-dimensional nilmanifold (Q1886721)

Let \(\mathcal{H}\) be the 3-dimensional Heisenberg group, which consists of all \(3\times 3\) real upper triangular matrices with diagonal entries 1, and \(\Gamma _1\) the discrete subgroup of \(\mathcal{H}\) consisting of all integral matrices. The authors call \(\mathcal{N}=\mathcal{H}/\Gamma _1\) the standard nilmanifold. They prove that if a finite group acts freely on the standard nilmanifold then the group is cyclic. Further they compose a list of such cylic groups.