I'm trying to understand the concept of free groups , and from what I've learned 
so far , a group $G$ is called a free group , if there is a subset $S ⊂ G$ such that
any element of G can be written uniquely as a product of elements of S , and their
inverses .

So , is $\Bbb{6Z}$ a free group ? or $\Bbb{Z}/17\Bbb{Z}$ ? or the group 

$$A = \left\langle\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix} , \begin{pmatrix} 0 & 2 \\ -1 & 0\end{pmatrix}\right\rangle$$ 

Let's take for example $\Bbb{18Z}$ (the coset $18k , k \in \Bbb{Z}$) , which is a subset of $\Bbb{6Z}$ (the coset $6k , k \in \Bbb{Z}$) . 

So , by the definition of free groups , do I need to take for example an element of $\Bbb{6Z}$ , e.g. $12$ , and few elements of $\Bbb{18Z}$ , and build (with them and their inverses) the element $12$ ?