**5.7 Examples: motion groups**

The first purpose of this section is to describe the dual space topology for some motion groups to which Theorem 5.58 applies easily. Afterwards, we use Theorem 5.58 to deduce a necessary and sufficient condition for a general motion group to have a Hausdorff dual space.

**Example 5.59** Let *G* = ℝ^{n} *SO*(*n*), the group of rigid motions of ℝ^{n}, *n* ≥ 2, and let *H* = ℝ^{n} *SO*(*n* – 1), where *SO*(*n* – 1) is identified with the stabilizer group of the vector (1, 0, …, 0) in *SO*(*n*). For *r* > 0 let *χ _{r}* denote the character of ℝ

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