Discrete Degree of Symmetry of Manifolds

We define the discrete degree of symmetry disc-sym(X) of a closed n-manifold X as the biggest \(m\ge 0\) such that X supports an effective action of \((\mathbb {Z}/r)^m\) for arbitrarily big values of r. We prove that if X is connected then disc-sym\((X)\le 3n/2\). We propose the question of whether for every closed connected n-manifold X the inequality disc-sym\((X)\le n\) holds true, and whether the only closed connected n-manifold X for which disc-sym(X)\(=n\) is the torus \(T^n\). We prove partial results providing evidence for an affirmative answer to this question.