free groups, finitely generated groups

I have three problems I wasn't able to solve by myself. :(

1) Is ( ) finitely generated group? Is it free?

2) Is free Abelian group?

3) Is free group?

I know that in order to check whether a group is finitely generated, I have to find a finite set that generates it, or prove it doesn't exist, but I can't do either. :(

Also, to check whether something is a free group, I would have to find its basis.

For example, is the basis of ?

I'm really lost, please help.