  We prove that the genus of a finite-dimensional division algebra is finite
whenever the center is a finitely generated field of any characteristic. We
also discuss potential applications of our method to other problems, including
the finiteness of the genus of simple algebraic groups of type $\textsf{G}_2$.
These applications involve the double cosets of adele groups of algebraic
groups over arbitrary finitely generated fields: while over number fields these
double cosets are associated with the class numbers of algebraic groups and
hence have been actively analyzed, similar question over more general fields
seem to come up for the first time. In the Appendix, we link the double cosets
with $\check{\rm C}$ech cohomology and indicate connections between certain
finiteness properties involving double cosets (Condition (T)) and Bass's
finiteness conjecture in $K$-theory.
