Find the conjugacy classes of $(123), (132) \in S_3$

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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

$$\{\sigma\tau\sigma^{-1}\in S_3\mid \sigma \in S_3\}.$$

In particular, then, for example, we have the conjugacy class of $(123)$ is $$\{(123)^e, (123)^{(12)}, (123)^{(13)}, (123)^{(23)}, (123)^{(123)}, (123)^{(132)}\},$$ which is $$\{ (123), (213), (321), (132), (123), (312)\}=\{(123), (132)\} .$$

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