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We know that E(n) is the isometry group of euclidean space. But considering E(n) or SE(n) or SO(n) itself as a Lie group with a left/right invariant Riemannian metric, what is the isometry group of E(n),SE(n) and SO(n)?

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The isometry group of $SO(n)$ (with the metric induced by the spectrum, e.g., the operator norm) is well-known. It is given by all $X\mapsto O'OX^{\pm 1}O^{-1}$, $O,O'\in SO(n)$ unless $n=4$, in which case we need to extend by $C_2$, swapping (1,4) and (2,3), and (4,1) and (3,2) entries in $\mathfrak{so}(4)$.

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  • $\begingroup$ I am not familiar with the concept of operator norm but is that same as the induced left or right invariant Riemannian metric on SO(n) as well? Can you mention some resources for operator norms and proving what you just answered? And how about SE(n)? What is the meaning of $C_2$ $\endgroup$ – Rama Seshan Chandrasekaran Jun 5 at 6:40

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