Thread: Free groups and subgroups

Let be the free group on two generators . Prove that the free group on 3 generators is a subgroup of .

Not completely sure how to go about doing this. So . Can I show that z can be written as the the product of 's and 's and their inverses?
Or is there some other method I should try?

well you won't actually find three letters in an alphabet that only has two.

try this instead: prove that the subgroup generated by x²,xy and y² is isomorphic to F₃.

So considering the subgroup , and ,
I'm trying the function f, where , can I guarantee this will be a homomorphism?

it seems pretty obvious it will be, we just need to check on all 9 products of the generators, right? (once you try to do this, i think it will become clear).

the real question is: is this an ISOMORPHISM?

find three letters in an alphabet that only has two.




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