I. Hambleton, L. Taylor and B. Williams conjectured a general formula in spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group $G$ and noetherian ring $R.$ The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group $S_5$, but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group $\mathrm{ SL}(2,\mathbb{ F}_3)$ is also a counterexample to the conjectured HTW-decomposition. Furthermore, we prove that for any finite group $G$ the rank of $G_1(\mathbb{ Z}G)$ does not exceed the rank of the expression in the HTW-decomposition.
Categories: math.AT math.GR