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I believe that group of rigid body motion is not compact.
I mean all transformations in $R^3$ that preserve distance.
But I need to know how to proof it?
From where I should start to prove it?

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up vote 3 down vote accepted

Let $G$ be the group of rigid motions over $\mathbb{R}^3$. Topologize $G$ in any ways you like. As long as $G$ remains to be a topological transformation group over $\mathbb{R}^3$, the evaluation map at origin. i.e. the map defined by

$$G \ni g \quad\mapsto\quad g(\vec{0}) \in \mathbb{R}^3$$ is a continuous surjection. If $G$ is compact, so does its image $\mathbb{R}^3$. Since $\mathbb{R}^3$ isn't compact, neither do $G$.

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