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?

Re: Free groups and subgroups

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

try this instead: prove that the subgroup generated by x^{2},xy and y^{2} is isomorphic to F_{3}.

Re: Free groups and subgroups

So considering the subgroup , and ,

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

Re: Free groups and subgroups

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?

Re: Free groups and subgroups

find three letters in an alphabet that only has two.

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