# Thread: Free groups and subgroups

1. ## Free groups and subgroups

Let $F_2$ be the free group on two generators $F_2=$. Prove that the free group on 3 generators $F_3$ is a subgroup of $F_2$.

Not completely sure how to go about doing this. So $F_3=$. Can I show that z can be written as the the product of $x$'s and $y$'s and their inverses?
Or is there some other method I should try?

2. ## 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 x2,xy and y2 is isomorphic to F3.

3. ## Re: Free groups and subgroups

So considering the subgroup $H=$, and $F_3=$,
I'm trying the function f, where $f(x^2)=a, f(y^2)=b, f(xy)=c$, can I guarantee this will be a homomorphism?

4. ## 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?

5. ## Re: Free groups and subgroups

find three letters in an alphabet that only has two.

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