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