# Conjugates of elements in a group

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• Feb 1st 2010, 05:25 AM
poorna
Conjugates of elements in a group
Hey, How do I find the conjugate classes of a dihedral group of order 2n?

And in Sn what is the number of r cycles? I thought it was nCr, but I have been asked to prove it is n!/(r((n-r)!)) ?
• Feb 1st 2010, 05:32 AM
poorna
Oh wait, that r cycles question..

I can choose elements in nCr ways.
Out of these I fix the least element in the first place. So that there are (r-1) ways to fill the 2nd place and so on. i.e. the cycle say (5,2,7) is replaced by (2,7,5). So ther are (r-1)! ways of doing this. So the total no. of ways is nCr * (r-1)! which is the reqd. nswer.

Am I right?
• Feb 2nd 2010, 03:17 AM
Swlabr
Quote:

Originally Posted by poorna
Hey, How do I find the conjugate classes of a dihedral group of order 2n?

What do elements of $D_{2n}$ look like? So, take an arbitrary element in this form, and conjugate it by another arbitrary element, $y^{-1}xy$. What does this element look like?

Such elements form your conjugacy class.