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.