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Do permutations such as those in the group $S_3$ move elements based on place (of elements in the input) or index?

E.g. does

$$\bigg(\frac{123}{231}\bigg)$$

move 1 to 2's place (e.g. if the input is 132, then the above permutation would result in 321) or to index 2 (in which case the input 132 would result in 312)?

Similary,

$$\bigg(\frac{123}{213}\bigg)°\bigg(\frac{123}{321}\bigg)=\bigg(\frac{123}{312}\bigg)$$

where reading from right to left the left permutation is seen to move (on input 321) 1 to the place of 2, 2 to the place of 1 and 3 to the place of 3. So this works on places.

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  • $\begingroup$ I don't understand the question. But if $\phi$ is that permutation then $\phi(1)=2$, $\phi(2)=3$ and $\phi(3)=1$. $\endgroup$ – David C. Ullrich Jun 4 '16 at 17:07
  • $\begingroup$ @DavidC.Ullrich So does $\phi$ mean that whereever element 1 is, it becomes 2 and whereever element 2 is it becomes 3? So it works by determining where the elements lie (places) rather than moving whatever is in index 1 to index 2 and index 2 to index 3 ... $\endgroup$ – mavavilj Jun 4 '16 at 17:11
  • $\begingroup$ @mavavilj permutations don't operate in terms of elements, just positions. Think about what would be the output if the input was, say, 456 instead of 132. $\endgroup$ – cronos2 Jun 4 '16 at 17:14
  • $\begingroup$ @cronos2 Well that contradicts my lecture notes. Here the above permutation is shown to move 1 to the place of 2 (wherever it is) and 2 to the place of 3 and so on. So 132 becomes 321. $\endgroup$ – mavavilj Jun 4 '16 at 17:16
  • $\begingroup$ No, it does not mean that. A permutation does move things from one place to another. A permutation is a function. If I said $f(x)=x^2$ would you ask whether that means that "wherever $2$ is, it becomes $4$"? No, that makes no sense. $\endgroup$ – David C. Ullrich Jun 4 '16 at 17:20

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