Group Theory

  • Thread starter PhysKid24
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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 dont 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

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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.
 
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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?
 

Dr Transport

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point groups are Abelian if memory serves me correctly.
 

matt grime

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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}
 
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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.
 

matt grime

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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.
 
Last edited:
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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.
 
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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 dont even know if I'm on the right track.
 

matt grime

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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

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

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