Dihedral group D5 - Symmetry of a Pentagon - Conjugacy classes

In summary, the conversation is discussing the conjugacy classes of D5, a dihedral group with 10 elements. The questioner is trying to understand how to determine the number of conjugacy classes, which should be 4 according to sources, but they can only identify 3 symmetries in a pentagon. The group theory expert explains that there is an additional class, (21345), which is equivalent to permuting (12) and then rotating (12345). They also mention that D5 is generated by two elements and that the specific representation in S5 is not necessary to determine the equivalence classes. The questioner then clarifies their understanding and the expert confirms that (13524) is a possible rotation in
  • #1
IanC89
3
0
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?
 
Last edited by a moderator:
Physics news on Phys.org
  • #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.)
 
  • #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).
 
  • #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.
 
  • #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.
 
  • #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.
 

1. What is the Dihedral group D5?

The Dihedral group D5, also known as the symmetry group of a regular pentagon, is a mathematical concept that represents all possible symmetries of a regular pentagon. It is a finite group of 10 elements, consisting of rotations and reflections of the pentagon.

2. How is the Dihedral group D5 represented?

The Dihedral group D5 can be represented using a Cayley table, which is a square matrix that shows the result of combining any two elements in the group. It can also be represented using a cycle notation, where each element is written as a product of cycles.

3. What are conjugacy classes in the Dihedral group D5?

Conjugacy classes in the Dihedral group D5 are subsets of elements that are considered equivalent under the operation of conjugation. In simpler terms, elements in the same conjugacy class are related to each other by a similarity transformation.

4. How many conjugacy classes are there in the Dihedral group D5?

There are 5 conjugacy classes in the Dihedral group D5. These include the identity element, 4 rotations, and 4 reflections. Each conjugacy class has a unique number of elements, with the identity element having only one element and the rotations and reflections having 2 elements each.

5. What is the significance of conjugacy classes in the Dihedral group D5?

Conjugacy classes in the Dihedral group D5 help to identify the underlying symmetries of the pentagon. They also play a crucial role in understanding the structure and properties of the group, as well as in solving mathematical problems related to the group.

Similar threads

I Determining the number of conjugacy classes in a p-group
    • Linear and Abstract Algebra
Replies
6
Views
1K
MHB Group Ring Integral dihedral group with order 6
    • Linear and Abstract Algebra
Replies
1
Views
1K
Subgroups of Symmetric and Dihedral groups
    • Linear and Abstract Algebra
Replies
1
Views
1K
Finiding a representative of the 2nd conjugacy class in A_n
    • Linear and Abstract Algebra
Replies
2
Views
2K
MHB Question on subgroup and order of the elements
    • Linear and Abstract Algebra
Replies
7
Views
1K
MHB Pinwheel Symmetry Formulas Exposed
    • Linear and Abstract Algebra
Replies
5
Views
2K
MHB Subgroups of the dihedral group D6
    • Linear and Abstract Algebra
Replies
9
Views
10K
MHB Statements about group-order-centralizer
    • Linear and Abstract Algebra
Replies
1
Views
643
Klein Four Group's action
    • Linear and Abstract Algebra
Replies
4
Views
1K
What are the conjugacy classes of subgroups of Z X Z?
    • Linear and Abstract Algebra
Replies
6
Views
6K

Hot Threads

Recent Insights

Back
Top