  In this paper, we show that for every abelian subgroup $H$ of a Garside
group, some conjugate $g^{-1}Hg$ consists of ultra summit elements and the
centralizer of $H$ is a finite index subgroup of the normalizer of $H$.
Combining with the results on translation numbers in Garside groups, we obtain
an easy proof of the algebraic flat torus theorem for Garside groups and solve
several algorithmic problems concerning abelian subgroups of Garside groups.
