Mathematicians have produced a wide variety of long and complex proofs of the existence of free groups, and there appears to be a strong emphasis upon finding better proofs that involve a variety of techniques. (Examples are here and "www.jstor.org/stable/2317030" [Broken] and here.) So clearly, mathematicians see free groups as very important objects. I don't understand why this is! Definition: Let X be a set. A group F is free on X if there is an (inclusion) map ψ: X → F and for ANY group G and (inclusion) map δ: X → G, there is a unique homomorphism of groups β: F → G such that δ=β°ψ. (I'm 80% sure inclusion maps are the relevant maps.) Why does it matter that for any given set, such an object exists? After all, given the wide variety of possible G's for any given X, F will have to be enormously complex, and so whatever additional structure a given G has that goes beyond the group-theory axioms, such structure will be wiped out in F if F is to bear homomorphisms to G's without those structures. Why would such a structureless object be of interest?