• Monobrow
In summary, the proposition states that if a group F is freely generated by a set U and also by a set V with the same number of elements, then F is also freely generated by V. This is supported by examples such as F_2 = <a,b> = <a,ab>, but it is unclear if this is always true. It seems that translating elements of U as words in V and stitching them back together could lead to a formal proof, but some technicalities may need to be addressed. It should also be noted that if the number of generators is infinite, not all sets with the same cardinality will be free generating sets and a free set may not generate the entire group.
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?

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...un by translating all the ui 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.

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.

## 1. What is a generating set for a free group?

A generating set for a free group is a subset of elements from the group that can be combined in various ways to generate all other elements of the group. Essentially, it is a set of building blocks that can create the entire group through multiplication and inversion.

## 2. How is a generating set different from a basis?

A generating set is similar to a basis in that both can be used to create all elements of a group. However, a generating set may have redundant elements, while a basis is a minimal set of elements that can still generate the entire group.

## 3. Can a generating set be infinite?

Yes, a generating set for a free group can be infinite. In fact, some free groups have infinitely many generators, such as the free group on countably infinite generators.

## 4. How do you determine the size of a generating set for a free group?

The size of a generating set for a free group is known as the rank of the group. It can be determined by finding the minimum number of elements needed to generate all elements of the group, which is equivalent to finding the size of a basis for the group.

## 5. What is the significance of generating sets in free groups?

Generating sets are important in free groups because they allow us to understand the structure of the group and how its elements can be combined. They also have applications in other areas of mathematics, such as algebraic topology and combinatorics.

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