Let denote the cyclic group of order n and let , assume is generated by the element r.

1. Prove that , and are cyclic subgroups of G of order 6.

Proof.

I think what is throwing me off is the notations here. Both H and J are cyclic since all of their elements are drew from a cyclic group (is this enough?).

Pick , we have since . The inverses, of course, would also have to be in there as well. Same thing for J.

Is this okay?

2. Let , prove that

I can tell that this is true, but I'm having trouble trying to show it is. Since , each one of those cyclic subgroups would constitute a coordinate here, but I'm confused in what to write to prove it.

Thanks!!!