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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.

mavavilj
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  • I don't understand the question. But if $\phi$ is that permutation then $\phi(1)=2$, $\phi(2)=3$ and $\phi(3)=1$. – David C. Ullrich Jun 04 '16 at 17:07
  • @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 ... – mavavilj Jun 04 '16 at 17:11
  • @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. – cronos2 Jun 04 '16 at 17:14
  • @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. – mavavilj Jun 04 '16 at 17:16
  • 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. – David C. Ullrich Jun 04 '16 at 17:20
  • @cronos2 No, that's simply totally wrong. If $\phi$ is the given permutation then an input of $456$ for $\phi$ simply makes no sense. – David C. Ullrich Jun 04 '16 at 17:21
  • Or not - if you're actually asking about what composition with a permutation does, that's different. – David C. Ullrich Jun 04 '16 at 17:25
  • @DavidC.Ullrich "Another notation that explicitly identifies the positions occupied by elements before and after application of a permutation [...]" http://mathworld.wolfram.com/Permutation.html – cronos2 Jun 04 '16 at 17:27
  • @DavidC.Ullrich I think the confusion here is the same that is described at https://gowers.wordpress.com/2011/10/16/permutations/ (and that led me to this question). If someone is still interested in that, I'd try to answer that question, and suffer the slings and arrows of outrageous downvotes if I get it wrong... – Marco13 Nov 12 '18 at 22:34

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