Find the conjugacy classes of $(123), (132) \in S_3$.
I could work out for {e} and [(12)]={(12),(13),(23)}. But I'm struggling to compute the conjugacy class for (123).. Can somebody point out any mistake?
- For (123)
- $(123)^{(123)} = (123)$
- $(123)^{(132)} = (123)$
- For (132)
- $(132)^{(123)} = (132)$
- $(132)^{(132)} = (132)$
So it seems that they are trivial classes…?
** I've gone through articles on maths SE and know the answer but I found that none of them actually showed the calculation for (123)…
Best Answer
Note that the conjugacy class of $\tau\in S_3$ is
$$\{\sigma\tau\sigma^{-1}\in S_3\mid \sigma \in S_3\}.$$