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I am currently taking an introductory course on Group Theory using the text "Groups and Symmetry" by M.A. Armstrong. In one of the exercises we are asked to show that the regular tetrahedron has 24 symmetries, if we allow products of reflections. I have successfully been able to list all of the permutations to do this, but the question is also asking about identifying a product of three reflections which exemplifies something that is neither a reflection nor a rotation. I am having trouble understanding how such a thing is possible, considering every time I attempt to compose three reflections I end up getting back a rotation (or at least something that appears to be a rotation and is already listed as one of the possible permutations).

It appears that a similar question has been asked here: Symmetry of tetrahedron that is not a reflection nor a rotation. Although, it doesn't seem that the question was explicitly answered. Any help with understanding this is much appreciated.

jjcluu
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    Label your vertices of the tetrahedron 1 2 3 and 4. Consider the symmetry that sends 1 to 2, 2 to 3, 3 to 4 and 4 to 1. – Rob Nicolaides Aug 31 '21 at 22:12
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    Thank you for your quick response. Would this not be four reflections though since we are sending vertices 1 to 2, 2 to 3, 3 to 4, and then 4 to 1? – jjcluu Aug 31 '21 at 22:21
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    Let $(ij)$ be the transposition of $i$ and $j$. Applied on $i$ it is $(ij)i=j$, on $j$ it is $(ij)j=i$, and on some $k\ne i,j$ it is $(ij)k=k$. Then consider the product $(12)(23)(34)$. Then $(12)(23)(34)1=(12)(23)1=(12)1=2$, $(12)(23)(34)2=(12)(23)2=(12)3=3$, $(12)(23)(34)3=(12)(23)4=(12)4=4$, and the last computation is done in the same spirit, or observe that $1$ is the missing value in the image. – dan_fulea Aug 31 '21 at 22:30
  • Thank you for the response @dan_fulea, that helps clarify what is going on. By doing this method, we end up with (2,3,4,1) as our symmetry, but how is this not equivalent to a reflection that we can obtain by composition of a reflection and a rotation or vice versa? – jjcluu Aug 31 '21 at 22:44
  • For example, if we perform (2,3,1,4)*(1,2,4,3) we obtain the same result. I guess that's mainly my confusion, since we can obtain the same symmetry by composing a rotation with a reflection. – jjcluu Aug 31 '21 at 22:47
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    I have posted a detailed answer to your question under the original question you linked. It uses the same example as @RobNicolaides suggested – David Sheard Aug 31 '21 at 23:34
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    @DavidSheard Thank you for the thorough response! Yes, I think I can see more clearly why the symmetries from RobNicolaides's example and my previous response are not the same symmetries given your answer. – jjcluu Sep 01 '21 at 00:06

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