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Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?

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I would start with Bob Oliver's Whitehead groups of finite groups. LMS Lecture Note Series, 132. Cambridge University Press, Cambridge, 1988. It's complicated and technical, but there are a bunch of examples computed in the last section that might answer your question. – Danny Ruberman Jul 24 '13 at 1:52
Ok. Thanks for the reference. – W. Politarczyk Jul 24 '13 at 13:41

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