  We define the discrete degree of symmetry $disc-sym(X)$ of a closed
$n$-manifold $X$ as the biggest $m\geq 0$ such that $X$ supports an effective
action of $({\mathbf Z}/r)^m$ for arbitrarily big values of $r$. We prove that
if $X$ is connected then $disc-sym(X)\leq 3n/2$. We propose the question of
whether for every closed connected $n$-manifold $X$ the inequality
$disc-sym(X)\leq n$ 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.
