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

1) Is ($\displaystyle \mathbb{R}^{*}, .$) finitely generated group? Is it free?

2) Is $\displaystyle \mathbb{Z}_3 \oplus \mathbb{Z}$ free Abelian group?

3) Is $\displaystyle \mathbb{Z}_6$ 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 $\displaystyle \{0, 1, 2, 3, 4, 5 \}$ the basis of $\displaystyle \mathbb{Z}_6$?

I'm really lost, please help.