Let $\Gamma$ be a discrete cocompact subgroup of the euclidean motion group $$ G={\mathbb R}^d\rtimes O(d). $$ Let $\phi:G\to O(d)$ the projection homomorphism. Is it true that $\phi(\Gamma)$ is finite?
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The answer is yes. This is a theorem of Bieberbach (see Corollary (8.26) of the book "Discrete Subgroups of Lie Groups" by M.S.Raghunathan (Springer Ergebnisse Tract). 

