$C'(1/6)$ groups' properties

Question

I need to prove the following properties of $C'(1/6)$ groups:

1. If $h^k$ equals relation, then $k$ is an order of $h$

2. All finite subgroups are cyclic

3. All abelian subgroups are cyclic

I guess I've proved 1) with Greendlinger's lemma, but I have no idea how to solve 2) - 3)

I'm solving some problems of a combinatorial group theory course, but I can't solve 2-3. I don't see any instruments except definition, Dehn's algorithm and Greendlinger's lemma. In the second, for example, I'm trying to explore structure of finite cyclic subgroup by applying Dehn's algorithm to generator and comparing it to relations using Greendlinger's lemma. It turns out (if I'm right) that square of generator must be a subword of some relation. But I don't see how that helps...

I would like to have a hint.