# Euclidean Conjugation group

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.

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## 1 Answer

What you need to prove is that

1. A conjugated reflection is always a reflection.

2. If you have two reflection then one is always a conjugate of the other.

It looks like you have made an approach to (1), though you're actually asserting more than you're proving it. How detailed a proof you need depends on what properties your text has already established.

Afterwards you need to prove (2). Again depending on the level of rigor expected of you, the proof could be something like: In order to make one reflection as a conjugate of the other, first rotate the plane so the desired reflection's centerline becomes parallel with that of the given one. Then translate such that the centerlines become equal, apply the given reflection, and finally translate and rotate backwards.

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Thanks..I do need a detailed proof, this is where I am stuck. – jan Nov 12 '12 at 0:26