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Let

$$ \sigma = (1\ 2\ 4), \tau =(1 \ 2 \ 5 \ 3) \ (4\ 7\ 6) $$

where $\sigma, \tau \in S_7$. Based on e.g. this answer, I think it should be the case that:

$$ \tau \sigma \tau^{-1} = (\tau(1)\ \tau(2)\ \tau(4)) = (2 \ 5 \ 7) $$

but when I work it out, I find that:

$$ \tau \sigma \tau^{-1} = (1\ 6\ 3) $$

and:

$$ \tau^{-1} \sigma \tau = (2\ 5\ 7) $$

but:

$$ (\tau^{-1}(1)\ \tau^{-1}(2)\ \tau^{-1}(4)) = (3\ 1\ 6) $$

Where am I going wrong here?

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1 Answer 1

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You have to be careful about whether you are combining your permutations from left-to-right or right-to-left, which corresponds to whether you are thinking of $S_n$ acting on $\{1,\dots,n\}$ on the right or left respectively.

To illustrate if we combine the permutations from left-to-right we get $$\tau\sigma\tau^{-1}=(1253)(476)(124)(1352)(467)=(163)$$ and $$\tau^{-1}\sigma\tau=(1352)(467)(124)(1253)(476)=(257).$$

However, if we combine from right-to-left we get $$\tau\sigma\tau^{-1}=(1253)(476)(124)(1352)(467)=(257)$$ and $$\tau^{-1}\sigma\tau=(1352)(467)(124)(1253)(476)=(163).$$

Usually if you call $\tau\sigma\tau^{-1}$ "the conjugate of $\sigma$ by $\tau$" you are imagining $S_n$ acting on the left and so you get the answer $(257)$ as stated in the answer you reference. I suspect that you are tacitly using the opposite convention which explains the discrepancy.

Note that your final equation is consistent because as permutations $(163)=(316)$.

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    $\begingroup$ The last part of the first sentence is not correct. If you combine permutations left-to-right (i.e., $\sigma \tau$ means do $\sigma$, then $\tau$), then you are thinking of $S_n$ acting on $\{1, 2, \ldots, n\}$ on the right, since you want $i (\sigma \tau) = (i \sigma) \tau$. $\endgroup$
    – Ted
    Jun 26 at 21:37
  • $\begingroup$ @Ted Thank you, yes even when you know that the issue comes down to mixing up left and right actions, sometimes it's still hard not to mix up left and right actions. I think my answer is now corrected $\endgroup$ Jun 26 at 21:42

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