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Dihedral group D5 - Symmetry of a Pentagon - Conjugacy classes

  1. Apr 13, 2010 #1

    I am struggling to get my head fully around the conjugacy classes of D5.

    Everywhere I have looked seems to say that there are 4 irreducible representations of D5 which implies that there are 4 conjugacy classes. However, when examining the symmetry of the pentagon I am only able to see 3 symmetries, namely the identity, reflections through an axis from a vertex to the mid-point of the opposite side and a rotation of 2*pi/5. In terms of permutations of a pentagon with vertexes labelled 1,2,3,4,5 clockwise, this would be (identity), (23)(45) and (12345).

    http://mathworld.wolfram.com/DihedralGroupD5.html" [Broken] says that there are 4 conjugacy classes, but I cannot see what the extra one must be. Any light shed on this would be a great help for me as I do not have a huge amount of in depth knowledge about group theory but I have a basic understanding of other groups, but cannot figure this one out.


    Edit * I have been trying to put further thought into it and one possible reason I have thought of is if the reflections through different axes are defined as a rotation and then just a reflection. This isn't how I have done things for a triangle or a square though, but it would, I think, give an extra set of actions that would have a different order to just a reflection, or just a rotation where the order of a reflection would be 2 and the order of a rotation would be 5 but a rotation and then a flip would have an order of 10 to get back to the identity. Does this sound feasible or am I barking up the wrong tree?
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Apr 14, 2010 #2
    What about (21345)?

    This is conjugate to (12345) in [tex] S_5[/tex], but because [tex] D_5[/tex] has only even permutations with the representation you've selected, it isn't conjugate to (12345) in [tex] D_5[/tex].

    (Which I think is quite close to what you added.)
  4. Apr 15, 2010 #3
    Would (21345) be equivalent to permuting (12) and then rotating (12345)? As I can see that only certain permutations of S5 are within D5 but are you saying that (21345) would be an additional class for D5? As I have come to the conclusion that the extra class is the class of (13254), I think (ie a flip and then a rotation, or a rotation and then a flip, as I added in the edit).
  5. Apr 16, 2010 #4
    Why bother with the specific representation inside S5? D5 is generated by 2 elements, s and r, with sr^-1 = rs, and o (r) = 5, o (s) = 2. That's enough to determine its equivalence classes.
  6. Apr 17, 2010 #5
    Sorry (21345) would be doing something very strange with the pentagon - I hope this didn't confuse you further.

    Try (13524) instead. This is physically possible and can only be obtained from (12345) by conjugation with an odd permutation.
  7. Apr 17, 2010 #6
    It's actually a rotation by twice the minimal angle. A flip followed by a rotation (or vice versa) is always just a flip about a different axis.
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