Free groups on some groups of same cardinality. As we all know free groups are always infinite, as they have no restrictions on their base sets. I have confusion as $S_3$ and $\mathbb{Z}_6$ both are groups of order $6$. What can we say about the free groups generated by these groups?
Are the isomorphic?
If they are not isomorphic, then what is the reason behind all that? 
I have studied so many questions on this site about free groups but didn't get any question like this. Someone told me not to ask such question here, but I am asking because this is my level of algebra. I can't understand that,s why I am asking. I am studying Combinatorial group theory by Wilhelm Magnus, Abraham Karrass and Donald Solitar and I came across such confusion. Please help me out to short out this problem. 

Yes they are isomorphic. When we say that $S$ is a generating set of a free group we just mean that it is a set of symbols which have inverses.
