Finding Conjugacy Classes in D5 Group

[PhysKid24]

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

 

[matt grime]

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.

 

[ComputerGeek]

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?

 

[Dr Transport]

point groups are Abelian if memory serves me correctly.

 

[matt grime]

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}

 

[ComputerGeek]

wow... some how I thought you wrote:

b = ba^{-2}

Don't know where that came from

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.

 

[matt grime]

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.

 

[Oxymoron]

Matt, can I ask a question while you are here?

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

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

 

[Oxymoron]

Continuing on from what I said.

if I consider the case where $$n$$ is even. Then obviously

$$a^n = e$$
$$b^2 = e$$
$$bab^{-1} = a^{-1}$$

and for $$n$$ odd, is it different? I don't even know if I'm on the right track.

 

[matt grime]

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.

 

[mathwonk]

you might also look at the geomketric picture of this group, i.e. its action on a polygon of n sides.