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The first proof of nonexistence is: [SS] S. Świerczkowski, On a free group of rotations of the Euclidean space, Indag. Math. 20 (1958) 376\[Dash]378. Problem first posed: [HS] H. Steinhaus, Problem 175, Coll. Math., 4 (1957) 243. I now have a demo on...