Hi guys,
I need help understanding cycle notation. Can you work me through this example please?
Kind regards,
Banana
Begin by "going" with the numbers beginning with 1: 1 goes to 2, 2 goes to 3, 3 goes to 1: we closed a first cycle (123); 4 goes to 4 and we don't write it, and then 5 goes to 6 and 6 goes to 5 ==> another cyce (56), so finally $\displaystyle \sigma=(123)(56)$ , and you have to read this as "1 -> 2 -> 3->1, 5->6->5.
Note that we didn't write the cycle (4): 4 goes to 4. This can be confusing sometimes if we don't know what n in $\displaystyle S_n$ are we working with, but in most cases we do so if we don't see a one-number cycle it means the permutation maps it to itself.
Now you try with $\displaystyle \pi$ (you've got to get one single 4-cycle)
Finally a warning word: when you multiply cycles, most authors choose to do it from right to left, as in functions composition.
Tonio