I'm working on a midterm review sheet for my MATH 417 class. It is noted that the midterm (which will be held at 10am Monday) will look very much like the review. I'm having a lot of trouble solving one question:
"Let b be a cycle of length at least 3 in Sn. Prove that b2 is a cycle if and only if the length of b is odd."
Any ideas?
Instead of writing out a full proof I think it is easiet to see what is going on between the parity by writing out a few example. First, we will consider when the length of is even:
If then .
If then .
If then .
Now look what happens when is odd:
If then .
If then .
If then .
It should be clear now how parity plays a role. When we have even number of elements in the cycle then the cycle terminates in the middle and returns to begining and that is why we get two disjoint cycles of halved length. And if we have an odd number of elements then the cylce manages to fully complete itself.