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An open question is whether the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow\widetilde{K_0 }\mathbb{Q} G$ in reduced $K$-theory from the integral to therational group ring is trivial for any group $G$. We will show that this isfalse, with a counterexample given by the group $QD_{32} *_{Q_{16}} QD_{32}$.We will also show how to compute the image of the map $\widetilde{K_0}\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ using representationtheoretic means, assuming $G$ satisfies the Farrell-Jones conjecture.

The passage from the integral to the rational group ring in algebraic $K$-theory