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Wiki says that the euclidean group: http://en.wikipedia.org/wiki/Euclidean_group is a topological group. Can you explain me what is the topology we take on it?

Thanks !

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Topologically the euclidean group is the product $O_n(\mathbb R)\times \mathbb R^n$ (but beware that the group structures are different: the euclidean group is a semi-direct product of $O_n(\mathbb R)$ and $\mathbb R^n$, not the direct product.)

Of course $ \mathbb R^n$ has its usual metric structure and $O_n(\mathbb R)$ has the induced topology from the inclusion $O_n(\mathbb R)\subset M_n(\mathbb R)\cong \mathbb R^{n^2}$.

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    I think the embedding of the $n$-dimensional Euclidean group into $\operatorname{GL}_{n+1}(\mathbb{R})$ given by \begin{bmatrix} A & x \\ 0 & 1\end{bmatrix} sheds light on both the algebraic and the topological structure. Note also that the same semi-direct product decomposition of the group of isometries into $O(X) \ltimes X$ holds for every normed real vector space by a theorem of Mazur and Ulam, see [this very enjoyable note](http://www.helsinki.fi/~jvaisala/mazurulam.pdf) by Väisälä.2012-05-19