Finding Conjugacy Classes in D5 Group

In summary: That might give you some clues.In summary, the group D5 is an abelian group, and there are two conjugacy classes - one for even n, and one for odd n.
  • #1
PhysKid24
22
0
Hi. Can anyone help me figure out how to find the conjugacy classes for a certain group and the elements in each class. I'm looking at the dihedral group of degree 5 (D5). I found the 10 elements in the group, but I don't know how to get the conjugacy classes and the elements in them? Can anyone help? Thanks. In the group, a^5=e; b^2=e; ab=ba^-1
 
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  • #2
Do it. Take an element x, work out all its conjugates. Use the relations to help:

aba^{-1} = ba^{-2}

so b and ba^{-2} are conjugate. Rinse and repeat, There are two cases depending on n.
 
  • #3
matt grime said:
Do it. Take an element x, work out all its conjugates. Use the relations to help:

aba^{-1} = ba^{-2}

so b and ba^{-2} are conjugate. Rinse and repeat, There are two cases depending on n.

you are assuming the group is Abelian. Do you know for a fact that D5 in this situation is Abelian?
 
  • #4
point groups are Abelian if memory serves me correctly.
 
  • #5
I am most definitely NOT assuming the group is abelian.

HINT: G is abelian IFF aba^{-1}=b for all a and b. I think you'll find I wrote

aba^{-1} = ba^{-2}
 
  • #6
wow... some how I thought you wrote:

b = ba^{-2}

Don't know where that came from :-p

Onto another question though. How do you know that ba^{-2} is the conjugate of b from:

aba^{-1} = ba^{-2}

I am not seeing the steps between.
 
  • #7
It is a dihedral group with generators a and b satisfying a^n=e=b^2 ab=ba^{-1], (n=5 for this particular example).

If you don't see why b and ba^{-2} are conjugate then this implies in my mind that you do not know what conjugate means.

b=ba^{-2} simply implies that a^2=e, that is all, by the way, nothing to do with abelian or otherwise.
 
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  • #8
Matt, can I ask a question while you are here?

On the subject of Dihedral groups, I am considering [tex]\mathcal{D}_n[/tex], the dihedral group of order [tex]2n[/tex].

How would I go about finding the normal subgroups of [tex]\mathcal{D}_n[/tex]. Do I consider the two distinct cases separately? That is, first let [tex]n=even[/tex] then work out the rotation maps [tex]a[/tex] and relfection maps [tex]b[/tex], and then let [tex]n=odd[/tex] and do the same thing?

What kind of things should I recognize (if any)? And will I be surprised?

I am just starting to think about these things, so I have no idea what to expect.

Thanks for any insight.
 
  • #9
Continuing on from what I said.

if I consider the case where [tex]n[/tex] is even. Then obviously

[tex]a^n = e[/tex]
[tex]b^2 = e[/tex]
[tex]bab^{-1} = a^{-1}[/tex]

and for [tex]n[/tex] odd, is it different? I don't even know if I'm on the right track.
 
  • #10
A simple test for normality of a subgroup:

N is normal in G if and only if N is a union of conjugacy classes (this is one proof that A_5 is simple)

Your second post doesn't seem related to the first. The relations defining the dihedral group say nothing about whether n is even or odd.
 
  • #11
you might also look at the geomketric picture of this group, i.e. its action on a polygon of n sides.
 

1. What is the D5 group?

The D5 group, also known as the dihedral group of order 10, is a mathematical group that represents the symmetries of a regular pentagon. It has 10 elements, including rotations and reflections.

2. How do you find conjugacy classes in D5 group?

To find the conjugacy classes in D5 group, we first need to identify the elements of the group. These include the identity element, 5 rotations, and 4 reflections. Then, we use the conjugacy equation to determine which elements are in the same conjugacy class.

3. What is the conjugacy equation for D5 group?

The conjugacy equation for D5 group is: g^-1hg, where g and h are elements of the group and g^-1 is the inverse of g. This equation helps us determine which elements are in the same conjugacy class.

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

There are 5 conjugacy classes in D5 group. These include the identity element, 4 rotations, and 4 reflections. This can be seen by using the conjugacy equation and identifying which elements have the same result.

5. Why is finding conjugacy classes important in D5 group?

Finding conjugacy classes in D5 group is important because it helps us understand the structure and properties of the group. It also allows us to simplify calculations and make predictions about the behavior of the group. Additionally, conjugacy classes have applications in various fields such as chemistry, physics, and computer science.

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