Having a tad bit trying to prove this question,
Show that the set of reflections is a complete conjugacy class in the euclidean group E. Also, do the same for the set of half-turns and inversions.
For the first set (reflections), I have that the set of reflections is a complete conjugacy class in E because the conjugates of a reflection are reflections with a translated, rotated and reflected mirror plane. Hence, the conjugate closure of a singleton containing a reflection is the whole E group. (This I got from some definitions). I don't know quite how to prove the question. Any help would be great or any suggestions. Thanks in advance.