Free group on a set B. If the free group is finitely generated, then B is finite. Let B be a set. For simplicity, assume B contains all the formal inverses of its elements. Let W(B) be the set of words created from elements in B, and let F(B) be the set of equivalence classes [w] where w is a word in W(B). Any two words in the same equivalence class are related by a finite number reductions (insertions or deletions) by elements in B.
Here's my question: If F(B) is finitely generated, how do we prove that B is finite? In another problem I proved that F(B) is freely generated by the set of equivalence classes of elements arising from B; that is, equivalence classes of the form [b] where b is an element of B. Call this set S(B). My math prof. told me to use the universal extension property (where each map from S(B) to a group G extends to a unique homomorphism from F(B) to G), but I'm not sure how to do it.

Hint: Assume $F(B)$ is generated by the words $w_1,\dots,w_n$, and take the set of letters occuring in these words $w_i$.
