TITLE: Intuition of characteristic property of the free group
QUESTION [5 upvotes]: Here is a theorem about characteristic property of the free group:

Theorem (Lee TM). Let $S$ be a set. For
any group $H$ and any map $f:S\to H$, there exists a unique homomorphism
$g:F(S)\to H$ extending $f$.

Here $F(S)$ is free group on $S$.
I know what it says but I don't know why it should be useful. i.e. What is the strategy of such theorems? How it can help to understand $F(S)$?
Can anyone enlighten it by a simple example?

REPLY [7 votes]: This characterisation emphasises that free groups are related to homomorphisms. Here is a simple proof which uses this characterisation:
Theorem. If $|X|\geq2$ then $F(X)$ is non-soluble.
Proof. The symmetric group $S_5$ can be generated by two elements. Therefore, by this characterisation of free groups, $F(X)$ subjects onto $S_5$ for all sets $X$ with at least two elements. As $S_5$ is non-soluble, as as solubility is preserved under homomorphic images, the result follows.
(Previously this answer used $S_3$ instead of $S_5$ to prove that free groups are non-nilpotent. The answer is now stronger, as non-soluble implies non-nilpotent.)