Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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.

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.