Let $S$ be any set and let $f:S \rightarrow F$ denote the free group on $S$. By definition, this means that for any group $X$ and any function $g:S \rightarrow X$ there exists a unique homomorphism $h:F \rightarrow X$ such that $h \circ f = g$.
1) How starting from this definition I do conclude that cannot have equations of the type $x^{3}=1;~xyx^{2}=1~;~\left( xy\right) ^{3}=1$ ?
2) Intuitively, the group $F$ to be free on $S$ it means that it is vectorial space over $\mathbb{Z}$, that is, every element of $F$ is a lineal combination on $\mathbb{Z}$ of elements of $S$?If not, then the one means what $F$ to be free on $S$?
3) Can anybody given an application of the following fact?
"every group is a quotient of a free group"