  Let $T(\Z_m \times \Z_n)$ denote the maximal number of points that can be
placed on an $m \times n$ discrete torus with "no three in a line," meaning no
three in a coset of a cyclic subgroup of $\Z_m \times \Z_n$. By proving upper
bounds and providing explicit constructions, for distinct primes $p$ and $q$,
we show that $T(\Z_p \times \Z_{p^2}) = 2p$ and $T(\Z_p \times \Z_{pq}) = p+1$.
Via Gr\"obner bases, we compute $T(\Z_m \times \Z_n)$ for $2 \leq m \leq 7$ and
$2 \leq n \leq 19$.
