Dihedral group D5 - Symmetry of a Pentagon - Conjugacy classes

[IanC89]

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?

 

[Martin Rattigan]

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.)

 

[IanC89]

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).

 

[Werg22]

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.

 

[Martin Rattigan]

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.

 

[Martin Rattigan]

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.