[Math] Finding conjugacy classes

group-theorypermutations

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:

  • (13)(12)(13)1=(13)(12)3=(13)3=1
  • (13)(12)(13)2=(13)(12)2=(13)1=3
  • (13)(12)(13)3=(13)(12)1=(13)2=2

So (13)(12)(13) leaves 1 fixed, and swaps 2 and 3.

Also, (12)(13) is not (123). Check:

  • (12)(13)1=(12)3=3
  • (12)(13)2=(12)2=1
  • (12)(13)3=(12)1=2

Thus, (12)(13) sends 1 to 3, and 3 to 2, and 2 to 1. So (12)(13)=(132), not (123).

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