1
$\begingroup$

If we have the following permutation: $\sigma = (2,1,4)(4,5,1,6) \in S_7$

Am I right in assuming it will permute in the following way?

$ \begin{array}{clcr} 1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \\ 6 \ 1 \ 3 \ 5 \ 2 \ 4 \ 7 \end{array}$

And, is there a quick way of finding a $\tau \in S_7$ such that $\tau^{-1} \sigma \tau = (1,2)(3,4,5)$, i.e. not having to write out all the permutations and checking which one fits? I've tried solving it with the cancellation law but screwed up somewhere.

Thanks.

  • 0
    Okay, thanks I will try this out. But basically $\tau$ is the important part, and $\tau^{-1}$ will just fall into place, so to speak? Thanks very much for your time and effort.2012-04-15

1 Answers 1

3

Your calculation is almost correct, but $5$ gets sent to $4$ and $6$ gets sent to $2$, so the final line should read $6 1 3 5 4 2 7$. Using this calculation, you can write $\sigma$ in disjoint cycle form: $\sigma = (1, 6, 2)(4, 5)$.

With this calculation in hand, can you find the $\tau$ you are looking for? Let me give you a start: $\tau (1, 2) \tau^{-1} = (\tau(1), \tau(2))$. (Think through why this is true.) You can similarly compute $\tau (3, 4, 5) \tau^{-1}$. Comparing with the expression for $\sigma$ will tell you how to choose $\tau$.

  • 0
    Yep, that's it. Since you're really doing composition of functions, the convention is to work from right-to-left.2012-04-15