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Understanding generating sets for free groups. 
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#1
Jan1213, 09:37 AM

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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? 


#2
Jan1413, 10:09 AM

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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 u_{1}u_{2}...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. 


#3
Jan1613, 05:59 AM

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P: 1,716

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.



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