Benjamin Crowell writes here, "The discontinuous transformations of spatial reflection and time reversal are not included in the definition of the Poincaré group, although they do preserve inner products."(adsbygoogle = window.adsbygoogle || []).push({});

http://www.lightandmatter.com/html_books/genrel/ch02/ch02.html

So, if I've understood this, the Poincaré group consists of all continuous transformations in Minkowski space that preserve the inner product: (1) translations and (2) the restricted Lorentz group (proper orthochronous Lorentz transformations in Minkowski space, i.e. rotations and boosts).

Is there a name for the corresponding set of transformations in Euclidean space? I gather the Euclidean group consists of (1) translations and (2) the orthogonal group (rotations and rotoreflections). It has a subgroup [tex]E^{+}\left ( n \right )[/tex] consisting of translations and the special orthogonal group (rotations). This [tex]E^{+}\left ( n \right )[/tex] is to Euclidean space what the Poincaré group is to Minkowski space, isn't it?

Is there a name or conventional symbol for the set of transformations in Minkowski space corresponding to the Euclidean group (translations together with the full Lorentz group), and does it form a group too?

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# Poincaré and Euclidean groups

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