Let a,b be elements of a group G. Given that (ab)^i = (a^i)(b^i) for two consecutive integers i. Show that G is not necessarily abelian. I'm am unable to find an counter example to show this.
Well, let's start with the smallest group that is not abelian.
That is S3, the permutations of 3 elements.
Now consider which values "i" are interesting.
For instance, i=1 is okay, since you'll get a trivial identity.
But for instance, i=2 is a problem, since abelian follows immediately (can you show that?).
What are other interesting values for "i" in the sense that you can say something immediately (as related tot S3)?