Timeline for What elements of my group am I missing and which group is it? [closed]
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| Apr 5, 2022 at 7:21 | history | closed |
John Rennie StephenG - Help Ukraine Jon Custer Davide Morgante DanielC |
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| Apr 4, 2022 at 11:39 | comment | added | god_operator | Thanks for the reply. However, if I found that $bc=cb$, then the group will be abelian. From $c^4=e$ and $b^2=e$, the elements of the group are the same elements as in $D_4$ (the groups are not the same though, since $D_4$ is not Abelian). I was asked to construct the character table, which would consist of 8, 1 dimensional irreps with 8 conjugacy classes, with a total of 64 elements in the character table. This seems to be very complicated for an exam question, so is there anywhere I went wrong? @Andrew | |
| Apr 3, 2022 at 16:56 | comment | added | Andrew | The reflection and rotation don't commute if the reflection is through the $xz$ or $yz$ plane. That is the case where $cb=bc^3$. The reflection and rotation do commute if the reflection is through the $xy$ plane. This is not difficult to prove. Put labels "a, b, c, d" on the corners of the top plane of the box, and "A, B, C, D" on the corners of the bottom plane of the box, and try doing a few examples. | |
| Apr 3, 2022 at 16:53 | comment | added | god_operator | you said that they commuted if the reflection was on the $xz$ or $yz$ plane. Do they commute if the reflection is on the $xy$ plane? Could you elaborate more on what elements would I find in the symmetry group if you know? @Andrew | |
| Apr 3, 2022 at 15:47 | review | Close votes | |||
| Apr 5, 2022 at 7:21 | |||||
| Apr 3, 2022 at 15:45 | comment | added | Andrew | Yes, I misunderstood your question and thought the reflection was through the $xz$ or $yz$ plane. Reflections through the $xy$ plane are easier. | |
| Apr 3, 2022 at 15:44 | comment | added | god_operator | But then if they indeed commuted, wouldn't the first relationship you mentioned in your comment $cb=bc^3$ not hold anymore? @Andrew | |
| Apr 3, 2022 at 15:41 | comment | added | Andrew | I think a reflection through the $xy$ plane should simply commute with rotations about the $z$ axis. | |
| Apr 3, 2022 at 15:31 | history | edited | John Rennie |
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| Apr 3, 2022 at 15:30 | comment | added | god_operator | sorry but i do not have enough score to post an image on Math SE @NiharKarve | |
| Apr 3, 2022 at 15:30 | comment | added | god_operator | I think you are refering to a reflection on a different plane from mine. If I do the operations you mentioned on my definition of reflection (see above I reflect over the $x-y$ plane), I do not get that the operations you said are equal. @Andrew | |
| Apr 3, 2022 at 15:22 | comment | added | Andrew | Just working out an example, it seems to me that $c b = b c^3$. In other words, looking at the square from the top, a reflection followed by a clockwise rotation, is equivalent to three clockwise rotations (or one counter clockwise rotation) followed by a reflection. | |
| Apr 3, 2022 at 15:20 | comment | added | Nihar Karve | Seems more like a question for Math.SE | |
| Apr 3, 2022 at 14:55 | history | asked | god_operator | CC BY-SA 4.0 |