1. Free groups and subgroups

Let $\displaystyle F_2$ be the free group on two generators $\displaystyle F_2=<x,y>$. Prove that the free group on 3 generators $\displaystyle F_3$ is a subgroup of $\displaystyle F_2$.

Not completely sure how to go about doing this. So$\displaystyle F_3=<x,y,z>$. Can I show that z can be written as the the product of $\displaystyle x$'s and $\displaystyle y$'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 $\displaystyle H=<x^2,xy,y^2>$, and $\displaystyle F_3=<a,b,c>$,
I'm trying the function f, where $\displaystyle f(x^2)=a, f(y^2)=b, f(xy)=c$, 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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