Understanding generating sets for free groups.

Monobrow

I was thinking about the following proposition that I think should be true, but I can't pove:

Suppose that F is a group freely generated by a set U and that F is also generated by a set V with |U| = |V|. Then F is also freely generated by V.

This is something that I intuitively think must be true when considering examples I have come across e.g( F_2 = <a,b> = <a,ab> with {a,b} and {a,ab} both free generating sets). Does anyone know if this is true?

CompuChip

I guess if F is generated by V, then you can write any element of U as a word in V in one and only one way.
So you can easily translate the word u₁u₂...u_n by translating all the u_i separately and stitching it back together. It feels like you are right that this should lead to a formal proof, though you may need to take care of some of the technicalities.

lavinia

note that if the numbr of generators is infinite then there are many generatin sets of the same cardinality but not all are free. Also a free set of elements of the same cardinality may not generate the whole group.