Class numbers and groups of algebraic groups

The class number of an algebraic group $G$ defined over a global field is the number of double cosets of the adele group $G_A$ with respect to the subgroups of integral and principal adeles. In most cases the set of double cosets has the natural structure of an abelian group, called the class group of $G$. In this article the class number of a semisimple group $G$ is computed, and it is proved that any finite abelian group can be realized as a class group.
Bibliography: 24 titles.