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I was wondering how I could simplify $((1 3 5)(2 3 4 1)(3 2 1))^{−1}$.

I was going to simplify the inner bracket section first then apply the inverse but I am having trouble simplifying it.

I see that $1\to3\to5\to1 $ but also $1\to3\to4\to1$ and$1\to3\to2\to1$. So I am not sure how to simplify it into one expression?

Thank you, if anyone could help that would be great :)

  • You should clarify / confirm whether your permutation "multiplication" is from right to left or from left to right as different authors use the same notation to mean different things. See for example this post. – JMoravitz May 07 '19 at 19:13
  • @JMoravitz good point. I didn't even think of that. – SlipEternal May 07 '19 at 19:14
  • As for reminders on how permutation multiplication works This post has a good explanation (noting that it follows the convention that permutations are applied from right to left so that $\sigma\cdot \pi = \sigma \circ \pi$, my personal preferred usage of the notation) – JMoravitz May 07 '19 at 19:16

1 Answers1

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$$\begin{align*}(135) \text{ takes }1 \to 3 \\ (2341)\text{ takes }3 \to 4 \\ (321) \text{ takes }4 \to 4\end{align*}$$

So, (135)(2341)(321) takes $1 \to 4$, so the inverse would take $4 \to 1$.

$$\begin{align*}(135)\text{ takes }2 \to 2 \\ (2341)\text{ takes }2 \to 3 \\ (321)\text{ takes }3 \to 2\end{align*}$$

So, (135)(2341)(321) takes $2 \to 2$, so the inverse would take $2\to 2$.

Etc.

Final result: $$(135)(2341)(321) = (1435)$$

$$((135)(2341)(321))^{-1} = (1534)$$

Edit: if the author does multiplication right to left:

$$\begin{align*}(321)\text{ takes }1 \to 3 \\ (2341)\text{ takes }3\to 4 \\ (135) \text{ takes }4 \to 4\end{align*}$$

etc.

Final Result:

$$(135)(2341)(321) = (1435)$$

$$((135)(2341)(321))^{-1} = (1534)$$

This problem happens to work out the same if it is left-to-right or right-to-left multiplication (not all problems will work out that way).

SlipEternal
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