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Group theory -- show H is a subgroup of O(2)

  1. Jan 18, 2017 #1
    1. The problem statement, all variables and given/known data
    Let ##R(\theta) = \left( \begin{array}{cc}
    \cos(\theta) & -\sin(\theta)\\
    \sin(\theta)& \cos(\theta)\\ \end{array} \right) \in O(2)## represent a rotation through angle ##\theta##, and

    ##X(\theta) = \left( \begin{array}{cc}
    \cos(\theta) & \sin(\theta)\\
    \sin(\theta)& -\cos(\theta)\\ \end{array} \right) \in O(2)## represent reflection around ##\frac{\theta}{2}##. Let m be a positive integer and H be the set such that ##H = \{R(\frac{2q\pi}{m}), X(\frac{2q\pi}{m}) | q = 0, 1, 2..., m-1\}##.

    Calculate ##R(\theta)R(\phi)##, ##R(\theta)X(\phi)## and ##X(\theta)X(\phi)##, express answers in terms of R and X. Show H forms a subgroup of ##O(2)##.

    2. Relevant equations


    3. The attempt at a solution
    ##R(\theta) R(\phi) =
    \left( \begin{array}{cc}
    \cos(\theta+\phi) & -\sin(\theta+\phi)\\
    \sin(\theta+\phi)& \cos(\theta+\phi)\\ \end{array} \right) = R(\theta + \phi)##

    ##R(\theta) X(\phi) =
    \left( \begin{array}{cc}
    \cos(\theta+\phi) & \sin(\theta+\phi)\\
    \sin(\theta+\phi)& -\cos(\theta+\phi)\\ \end{array} \right) = X(\theta + \phi)##

    ##R(\theta) X(\phi) = R(\theta - \phi)##

    I'm not really sure of the significance of these calculations, or if there is one. Does this have a geometric interpretation? I've been told that the set ##O(2)## is somehow related to the permutation of 3 points but haven't been able to find out why.
    To show it's a subgroup of O(2) I have to show ##R \times X \in O(2)## where ##\times## is matrix multiplication and ##a^{-1} \in H##. Do I need to use the argument ##\frac{2q\pi}{m}## in these calculations? So for example let ##a = R(\frac{2q\pi}{m})##. Any help is very much appreciated, thank you! :)

    I've gone ahead and attempted the problem with the argument above, and set a = R, b = X. I've already calculated RX to be ##= R(\theta - \phi)## above: ## =
    \left( \begin{array}{cc}
    \cos(0) & \sin(0)\\
    \sin(0)& -\cos(0)\\ \end{array} \right)
    =
    \left( \begin{array}{cc}
    1& 0\\
    0 & 1\\ \end{array} \right)##
    I wasn't really expecting to get the identity matrix, I suppose I show it's part of O(2) by showing ##A^T A = I ## and of course, it is a real 2x2 matrix.
     
    Last edited: Jan 18, 2017
  2. jcsd
  3. Jan 18, 2017 #2

    Orodruin

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    To prove that it is a subgroup, you have to show that it is a subset of O(2) that satisfies all the group axioms. That it is a subset of O(2) should be clear so you need to show that the group axioms are satisfied.
     
  4. Jan 18, 2017 #3
    By multiplying the two elements and showing they are part of ##O(2)## I'm not demonstrating closure then? I should actually show they are part of ##H##? Matrix multiplication is associative, the identity is ##
    \left( \begin{array}{cc}
    1& 0\\
    0 & 1\\ \end{array} \right)## and since ##RX = I ## the implication is that ## R = X^{-1}## and ##X = R^{-1}##, so the inverses both exist and are part of H. Does this demonstrate that the subset is a subgroup? Or group (not sure of the difference actually)? It certainly satisfies group axioms, so that makes it a subgroup?
     
  5. Jan 18, 2017 #4

    fresh_42

    Staff: Mentor

    This is only the closure of ##O(2)##, not of ##H##.
    Yes.
    Yes. But you also have to show ##R(\theta) \cdot X(\phi) \in H\, , \,R(\theta) \cdot R(\phi) \in H\; , \;X(\theta) \cdot X(\phi) \in H## and ##X(\phi) \cdot R(\theta) \in H## for all ##\theta, \phi##.
    For a subgroup, you need to show ##a\cdot b^{-1} \in H## for all ##a\, , \,b \in H##. Associativity is inherited. (Of course you can as well show ##a\cdot b \in H\, , \,a^{-1} \in H\; , \;1 \in H## separately but ##a\cdot b^{-1} \in H## does it all in one step.) However, it has to be shown for all ##a,b \in H##, which means all pairs ## (a,b) \in \{R(\theta),X(\phi)\}^2## in this case.

    I'm not sure about the connection to permutations you mentioned. (There is one for its tangent space but I don't know about the group itself. On the other hand it's full of symmetries as you have already calculated above.)
     
  6. Jan 18, 2017 #5

    Orodruin

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    Yes. If the subset is not closed under the multiplication, it is not a group of its own.

    What are your R and X here? The inverse of a rotation is not a reflection.
     
  7. Jan 18, 2017 #6
    Couldn't rotation through ##\theta## undo a reflection through ##\frac{\theta}{2}## in a few cases? I'm not at all sure how this whole thing relates to geometry. What exactly is being rotated or reflected? Or do these matrices just result in rotation or reflection when applied to other matrices or vectors?
     
  8. Jan 18, 2017 #7

    fresh_42

    Staff: Mentor

    Look in a mirror. Can you reverse the change of left and right by a rotation? Reflections change the orientation, rotations don't. Or in terms of the matrix group here: one has determinant ##-1##, the other on ##1##.
     
  9. Jan 18, 2017 #8

    Orodruin

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    Try taking the determinant of ##RX## ... The inverse of a rotation by ##\theta## is a rotation by ##-\theta## (or equivalently, by ##2\pi-\theta##).
     
  10. Jan 18, 2017 #9
    So ##RX## shouldn't actually be equal to ##I## then? I've probably made a mistake, I'll check. I suppose I was thinking of highly symmetric things like circles, which aren't really changed by these things, when I say rotation could undo reflection, but only in some few cases.
     
  11. Jan 18, 2017 #10
    But aren't both theta and phi defined in H as being ##\frac{2q\pi}{m}##? I'd shown that ##R(\theta)X(\phi) = R(\theta - \phi)##, so ##R(\frac{2q\pi}{m})X(\frac{2q\pi}{m}) = R(\frac{2q\pi}{m}- \frac{2q\pi}{m}) = R(0)##, which I thought would be
    ##R(0) = \left( \begin{array}{cc}
    1& 0\\
    0 & 1\\ \end{array} \right)##
    I'm also not completely sure what that last bit of notation means: ## (a,b) \in \{R(\theta),X(\phi)\}^2##
    Why is this bracket squared?
     
  12. Jan 18, 2017 #11

    fresh_42

    Staff: Mentor

    So? This doesn't change my argument, only the amount of possible angles.
    There might be a mistake in here. Check the determinants. And, yes, an equation like this would do the job. But you still have to cover the possibility of different angles ##\frac{2q\pi}{m}## and ##\frac{2p\pi}{m}## in ##R(\frac{2q\pi}{m})## and ##X(\frac{2p\pi}{m})##. And what happens with ##R \cdot R' \, , \, X \cdot X' \, , \, X\cdot R\,##? The latter because ##H## is eventually not Abelian. (You haven't checked this.)
    It means ## (a,b) \in \{R(\theta),X(\phi)\} \times \{R(\theta),X(\phi)\} = \{R(\theta),X(\phi)\}^2## or ##a,b \in \{R(\theta),X(\phi)\}## or ##(a \in \{R(\theta),X(\phi)\} ## and ## b \in \{R(\theta),X(\phi)\})##. These are significantly more cases as only ##RX##.
     
  13. Jan 18, 2017 #12
    Ah. I hadn't realised the angles could be different, I assumed it would always be ##\theta = \phi = \frac{2q\pi}{m}##, but I see why that's not the case. A shame, it made life that much easier. So what I've worked out is only for the case of ##\theta = \phi##. Back to the drawing board!

    I'll try that again and see where I get to. Your time and help is much appreciated, thank you! :)
     
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