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

Burnside problem for measure preserving groups of toral homeomorphisms and for 2-groups of toral homeomorphisms