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I think I can see where this is going, but I don't understand your notation so I will use different, but I hope that this will make sense.... The point here is that if translation is a symmetry operation (call it $t$) and rotation is a symmetry operation (call it $C_n$ - where $n$ is the number of $C_n$ operations required for a full turn - i.e. an $n-$fold ...


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Since you have not provided a direct reference, it's hard to be completely sure (and particularly to pin down the details), but there's really only one general idea that this can refer to. In general, any arbitrary isometry $S$ of euclidean space has the form $$ \mathbf x\mapsto R\mathbf x+\mathbf t, $$ where $\mathbf t$ is an arbitrary vector and ...



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