  A group $G$ is said to be periodic if for any $g\in G$ there exists a
positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic
group of homeomorphisms on the 2-torus that preserves a measure $\mu$ is
finite. Moreover if the group consists in homeomorphisms isotopic to the
identity, then it is abelian and acts freely on $\mathbb{T}^2$. In the
Appendix, we show that every finitely generated 2-group of toral homeomorphisms
is finite.
