2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62880Let $G$ 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.12 pages, 2 figures, AMSLaTeXRings and AlgebrasAlgebraic Topology19A31; 16S34; 16S40; 16E20; 16D90; 20J05text