Dihedral group D5 - Symmetry of a Pentagon - Conjugacy classes

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Discussion Overview

The discussion centers on the conjugacy classes of the dihedral group D5, particularly in relation to the symmetries of a pentagon. Participants explore the relationship between the number of irreducible representations and the perceived number of conjugacy classes, as well as the implications of different permutations within the group.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the number of conjugacy classes in D5, noting that they can only identify three symmetries: the identity, certain reflections, and a specific rotation.
  • Another participant points out that while (21345) is conjugate in S5, it is not conjugate in D5 due to the restriction to even permutations.
  • A different participant suggests that (21345) could be viewed as equivalent to a combination of a permutation and a rotation, while proposing that (13254) might represent an additional class.
  • One participant emphasizes that D5 is generated by two elements, which should be sufficient to determine its equivalence classes without needing to reference specific representations in S5.
  • Another participant clarifies that (13524) is a physically realizable permutation that can be derived from (12345) through conjugation with an odd permutation.
  • It is noted that a flip followed by a rotation results in a flip about a different axis, suggesting a relationship between these operations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the number of conjugacy classes in D5, with multiple competing views and interpretations of the permutations and their relationships within the group.

Contextual Notes

The discussion reflects varying interpretations of the symmetries and conjugacy classes, with some participants relying on specific representations and others focusing on the group structure itself. There is an acknowledgment of the complexity involved in understanding the relationships between different permutations and their classifications.

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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