2
$\begingroup$

I need to determine the missing number to fulfill the following reproduction:

$$\pi=\pmatrix{1&2&3&4&5&6&7&8&9\\3&5&9&4&1&2&6&7&8}$$

And here is the representation of $pi$ as the product of transpositions:

$$(1 i) (1 3) (2 5) (9 8) (8 7) (7 6) (3 6) = \pi$$

The result for $i$ must be 2 but I have no clue how to get there. I tried to start at $(3 6)$ and then check on which transposition it gets pictured on but it doesn't seem to work for me. Any ideas on how to generally solve these kind of problems?

-Freddy

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    Which way do you multiply transpositions? Does $(76)(36)$ send $7$ to $3$, or does it send $3$ to $7$?2012-11-04
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    thats a good question. apparently I learned that you send from the inner to the outer so 3 to 7 I suppose?2012-11-04
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    Note, there is no unique decomposition of this permutation into the product of transpositions, so I'm not sure what you mean by "here is *the* transposition"2012-11-04
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    Have you tried first expressing the permutation as a product of disjoint cycles? Or have you not encountered cycle notation? It would help to know if you were asked to express the permutation as the product of transpositions, or if you were given the transpositions and asked to find the missing number?2012-11-04
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    actually I was asked to find the missing number and the transposition was already given. We barely spoke about cycle notation due to lack of time...2012-11-04
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    Okay, thanks for clarifying. I just find the given transposition confusing, given the initial representation of $\pi$. As I mention in my comment, there are many ways to write a permutation as the product of transpositions.2012-11-04
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    okay I understand now, thank you both!2012-11-04

2 Answers 2