Let G be a group, and . For any positive integer we define by
Now prove by induction:
If , then
*edit: Please, no spoilers. Just a nudge is all I need. Thanks.
So we know ab=ba. The base case n=1 is trivial. We need to carry out the inductive step. The inductive hypothesis is that . We need to prove that . How can we see here something that appears in the inductive hypothesis? When we see it, we can use ab=ba (more than once).
the trick is to see how you're going to get from the "n-step" to the "n+1-step".
let's see if we can prove from ab = ba.
do you see it coming...? we can just switch the middle pair.
now try it for , and see if you can generalize.....