I've been having problems with finding conjugacy classes. I don't really understand how to do it properly.
Say we look at a S3 group: $S_3=]e, (12), (13), (23), (123), (132)]$
If we look at just $(12)$ first:
$e(12)e^{-1} = e(12)e=(12)$ I can understand this.
$(12)(12)(12)^{-1} = (12)(12)(12) = (12)$ Now here I'm a little confused. Wouldn't $(12)^{-1} = (21)$?
Then $(13)(12)(13)^{-1} = (13)(12)(13) = (32)$ how did that become $(32)$?
I thought you would start from the right so do $(12)(13)$ first then that would be $(123)$ then $(13)(123)$: 1 stays the same. 3 goes to 2? then 3 stays so (123)? I'm really confused
Best Answer
(12) and (21) are the same permutation.
As for (13)(12)(13), well, simply compute what it does to 1,2,3:
So (13)(12)(13) leaves 1 fixed, and swaps 2 and 3.
Also, (12)(13) is not (123). Check:
Thus, (12)(13) sends 1 to 3, and 3 to 2, and 2 to 1. So (12)(13)=(132), not (123).