  Let $G$ be a finitely generated abelian-by-finite group and $k$ a field of
characteristic $p\ge 0$. The Euler class $[k_G]$ of $G$ over $k$ is the class
of the trivial $kG$-module in the Grothendieck group $G_0(kG)$. We show that
$[k_G]$ has finite order if and only if every $p$-regular element of $G$ has
infinite centralizer in $G$. We also give a lower bound for the order of the
Euler class in terms of suitable finite subgroups of $G$. This lower bound is
derived from a more general result on finite-dimensional representations of
smash products of Hopf algebras.
