Thread: free group

1. free group

Hi,

How can I show that the free group with 2 generators is not solvable? After that how can I show that the only free group that is solvable is the group with one generator?

Thanks.

2. Re: free group

hmm...i think....that you should prove first that every homomorphic image of a solvable group is solvable. then find a group with two generators that isn't solvable.

3. Re: free group

if I choose the group with 2 generators to be ZxZ, is it solvable?

4. Re: free group

well, yeah, because ZxZ is abelian. but you're puting the cart before the horse. why is it sufficent to prove that a homomorphic image of a solvable group is solvable, and have you proven that yet?

5. Re: free group

For that I have a proof. But than I get that the free group with 2 generators is solvable, and I should prove that is not solvable, so there is something wrong in this...

6. Re: free group

ok, suppose you know the homomorphic image of a solvable group is solvable. suppose you find a group with 2 generators that ISN'T solvable.

isn't such a group a quotient group of the free group on two generators? and aren't quotient groups of a group just homomorphic images of that group

(the first isomorphism theorem)?