A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups G for which the integral group ring ℤG has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that ℤG has SFC provided at most one copy of the quaternions ℍ occurs in the Wedderburn decomposition of the real group ring ℝG. This generalises the Eichler condition in the case of integral group rings.

中文翻译：

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整群环上模的一个抵消定理

一个长期存在的问题，其根源于低维同伦理论，是对所有有限群进行分类G其中积分群环ℤG有稳定的免费取消（SFC）。我们通过给 SFC 一个条件来扩展 RG Swan 的结果，并用它来证明 ℤGSFC 是否提供了最多一份四元数 ℍ 出现在实群环的 Wedderburn 分解中 ℝG. 这概括了整群环情况下的 Eichler 条件。