0

I feel so lost on finding the order of a permutation. I understand the definition is the # of transpositions it can be "broken down" into, but how do I go about actually finding these transpositions?

For instance:

Find the order of (1 2 5)(3 4), (1 2 5)(3 4 7), and (1 2 4)(3 4 7 8 9).

Is there a "trick" to finding these orders? I have an exam next week on this material and I really need some help understanding it. Thanks!

EDIT: I have the solutions (6, 3, and 7), but I don't know how the book got to these answers.

Git Gud
  • 31,356
Max
  • 1,620
  • Yes, there's "trick". Try to find it. Here some hints regarding the first permutation. How many times do you have to act on $1$ in order to return back to $1$? What about $2$ and $5$? Same for $3$ and $4$. Combining this information, can you find the trick? Disclaimer: The trick I'm hinting to only works for disjoint cycles so you need to rewrite the last one as a disjoint product of cycles in order to use it. – Git Gud Sep 25 '16 at 13:42
  • 1 needs 3 "actions." 2 needs 3. 5 needs 3. Each element of a cycle needs "length-many" actions to get back to itself. But I don't see how to get the order of 6. – Max Sep 25 '16 at 13:48
  • Well, you know $3$ and $5$ need two actions each (or four, or six or any multiple of $2$). But in order to comply with the actions required for $(1,2,5)$ you need a multiple of $3$. Can you find the answer now? – Git Gud Sep 25 '16 at 13:50
  • OHH is it the lowest common multiple for the lengths of each cycles? – Max Sep 25 '16 at 13:54
  • It is $\ddot \smile$ But remember: disjoint cycles. – Git Gud Sep 25 '16 at 13:56
  • 1
    See: How to write permutations as product of disjoint cycles?, and also How to find the order of a permutation?. – amWhy Sep 25 '16 at 14:38

0 Answers0