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Group Theory

  1. Mar 3, 2005 #1
    Hi. Can anyone help me figure out how to find the conjugacy classes for a certain group and the elements in each class. I'm looking at the dihedral group of degree 5 (D5). I found the 10 elements in the group, but I dont know how to get the conjugacy classes and the elements in them? Can anyone help? Thanks. In the group, a^5=e; b^2=e; ab=ba^-1
  2. jcsd
  3. Mar 4, 2005 #2

    matt grime

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    Do it. Take an element x, work out all its conjugates. Use the relations to help:

    aba^{-1} = ba^{-2}

    so b and ba^{-2} are conjugate. Rinse and repeat, There are two cases depending on n.
  4. Mar 11, 2005 #3
    you are assuming the group is Abelian. Do you know for a fact that D5 in this situation is Abelian?
  5. Mar 11, 2005 #4

    Dr Transport

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    point groups are Abelian if memory serves me correctly.
  6. Mar 11, 2005 #5

    matt grime

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    I am most definitely NOT assuming the group is abelian.

    HINT: G is abelian IFF aba^{-1}=b for all a and b. I think you'll find I wrote

    aba^{-1} = ba^{-2}
  7. Mar 11, 2005 #6
    wow... some how I thought you wrote:

    b = ba^{-2}

    Don't know where that came from :-p

    Onto another question though. How do you know that ba^{-2} is the conjugate of b from:

    aba^{-1} = ba^{-2}

    I am not seeing the steps between.
  8. Mar 11, 2005 #7

    matt grime

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    It is a dihedral group with generators a and b satisfying a^n=e=b^2 ab=ba^{-1], (n=5 for this particular example).

    If you don't see why b and ba^{-2} are conjugate then this implies in my mind that you do not know what conjugate means.

    b=ba^{-2} simply implies that a^2=e, that is all, by the way, nothing to do with abelian or otherwise.
    Last edited: Mar 11, 2005
  9. Mar 22, 2005 #8
    Matt, can I ask a question while you are here?

    On the subject of Dihedral groups, I am considering [tex]\mathcal{D}_n[/tex], the dihedral group of order [tex]2n[/tex].

    How would I go about finding the normal subgroups of [tex]\mathcal{D}_n[/tex]. Do I consider the two distinct cases separately? That is, first let [tex]n=even[/tex] then work out the rotation maps [tex]a[/tex] and relfection maps [tex]b[/tex], and then let [tex]n=odd[/tex] and do the same thing?

    What kind of things should I recognize (if any)? And will I be surprised?

    I am just starting to think about these things, so I have no idea what to expect.

    Thanks for any insight.
  10. Mar 22, 2005 #9
    Continuing on from what I said.

    if I consider the case where [tex]n[/tex] is even. Then obviously

    [tex]a^n = e[/tex]
    [tex]b^2 = e[/tex]
    [tex]bab^{-1} = a^{-1}[/tex]

    and for [tex]n[/tex] odd, is it different? I dont even know if I'm on the right track.
  11. Mar 23, 2005 #10

    matt grime

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    A simple test for normality of a subgroup:

    N is normal in G if and only if N is a union of conjugacy classes (this is one proof that A_5 is simple)

    Your second post doesn't seem related to the first. The relations defining the dihedral group say nothing about whether n is even or odd.
  12. Mar 23, 2005 #11


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    you might also look at the geomketric picture of this group, i.e. its action on a polygon of n sides.
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