Dihedral group D5 - Symmetry of a Pentagon - Conjugacy classes

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SUMMARY

The discussion centers on the conjugacy classes of the dihedral group D5, which describes the symmetries of a regular pentagon. Participants confirm that D5 has four conjugacy classes despite the initial observation of only three symmetries: the identity, reflections, and a rotation of 2π/5. The additional class is identified as the permutations resulting from combinations of flips and rotations, specifically (13254) and (13524), which arise from conjugation with odd permutations. This highlights the complexity of understanding group representations within D5.

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  • Understanding of dihedral groups, specifically D5
  • Basic knowledge of group theory and conjugacy classes
  • Familiarity with permutations and their representations in symmetric groups
  • Concept of irreducible representations in group theory
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IanC89
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Hi

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" 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.

Thanks.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?
 
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What about (21345)?

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

(Which I think is quite close to what you added.)
 
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).
 
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.
 
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.
 
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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