Writing a Permutation as a product of Disjoint Cycles
Write the following as a product of disjoint cycles:
$(1 3 2 5 6)(2 3)(4 6 5 1 2)$
I know from my solutions guide that the answer is:
$(1 2 4)(3 5)(6)$
but I don't know how to do that. I started by writing it as a product of transpositions as such:
$(1 3)(3 2)(2 5)(5 6)(2 3)(4 6)(6 5)(5 1)(1 2)$
I want to put this in standard form, but I don't know where to go from here...If anyone could help shed light on the procedure to do this, it would be very helpful.
$\endgroup$2 Answers
$\begingroup$First write this product with one permutation and then into product of disjoint cycles:
$$(1 3 6)(1357) (1234)= \left(\begin{matrix}1&2&3&4&5&6\\ 2&4&5&1&3&6\end{matrix}\right)=(1 2 4)(3 5).$$
$\endgroup$ 3 $\begingroup$For each number from 1 to 6, figure out where the permutation takes it, and continue this for each one until you build all the cycles.
For example, we follow 1's path to 2, to 3, to 2, so 1 goes to 2. Starting at 2 now, we see 2 goes to 4, and that's it. Finally, 4 goes to 6, then 1, so we have completed a cycle. Continue this until all cycles are found.
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