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I have been asked

Does the set of all reflections in the plane form a group? Explain.

I thought that the answer was that it wouldn't as if I reflected the parabola $y=x^2$ through the y axis, I have a shape that is more of an "n" shape rather than a "u" shape.

However in my notes I have found that when listing the elements of a symmetry group of the same parabola, a reflection in the y axis exists.

Would someone be able to help clarify where I am going wrong? It could be with understanding the concept of a group, or what it means to reflect a function, or indeed both. Thanks for any assistance available

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    $\begingroup$ How have you had reflection defined? $\endgroup$ – Tobias Kildetoft Aug 13 '14 at 8:49
  • $\begingroup$ My understanding of a reflection was a transformation that retains the same shape. I am away from my notes now, so I could be mistaken, but I think it was somewhere along those lines. $\endgroup$ – Roy Sheehan Aug 13 '14 at 8:55
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    $\begingroup$ There is no way to do an exercise like this without the precise definition of the term "reflection". $\endgroup$ – Tobias Kildetoft Aug 13 '14 at 9:07
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    $\begingroup$ My understanding was that there are two types of isometries of $\mathbb{R}^n$: rotations, which preserve orientation, and reflections, which reverse orientation. If this is close to your definition, the identity map is not a reflection. $\endgroup$ – robjohn Aug 13 '14 at 9:18
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It depends on the group action. If the group action is composition (i.e. performing one reflection after another), then you have to answer four questions:

  1. Is the identity operation (i.e. leaving the plane as-is) representable as one of the reflections in the set?
  2. For any reflection $R$ you make, can you get back to the original function using another reflection (the inverse of $R$).
  3. If you perform two arbitrary reflections in the set, $R_1$ and $R_2$, one after another, can the net result always be represented as a single reflection $R$ in the set?
  4. Are compositions of reflections in the group associative?

If the answer to all questions is 'Yes', then the set of reflections you're looking at forms a group under composition.

Based on your initial description, assuming composition, the answer to your question is 'No', since leaving the plane as-is doesn't (typically) constitute a "reflection" and thus (1) is false.

As Dr. Kildetoft indicates, the answer depends on how exactly you define "reflection".

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  • $\begingroup$ @user1729 I was about to ask how you knew I had finished my PhD recently when I saw the edit comment that it was just to be on the safe side (personally, I don't care whether people call me dr. or mr, since I usually prefer to be called by my first name anyway). $\endgroup$ – Tobias Kildetoft Aug 13 '14 at 9:31
  • $\begingroup$ @Tobias Well, actually, I did do a quick google after my edit - congratulations! Did you just submit your hardbound copy recently? I found this page, dated yesterday... $\endgroup$ – user1729 Aug 13 '14 at 9:32
  • $\begingroup$ @user1729 No, I am not sure why that page was updated yesterday. I submitted April 1st and defended on the date listed there. I have not done anything in relation to it since. $\endgroup$ – Tobias Kildetoft Aug 13 '14 at 9:35
  • $\begingroup$ @Tobias So you had no corrections? Is that common on the continent (or do you have corrections before your defence)? Certainly, in the UK, noone escapes with less than "minor" (1-month) corrections. $\endgroup$ – user1729 Aug 13 '14 at 9:37
  • $\begingroup$ @user1729 The submitted thesis is what is evaluated. The only way I could be required to make corrections was if it was completely rejected, in which case I would have had 3 months to fix it (I think). Apparently, the system is somewhat different. That is not to say that I have not gotten a lot of good feedback that will make the resulting paper(s) better, but there is no requirement that I submit an updated thesis. $\endgroup$ – Tobias Kildetoft Aug 13 '14 at 9:40

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