free group

• Oct 23rd 2011, 05:35 AM
Veve
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.
• Oct 23rd 2011, 09:15 AM
Deveno
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.
• Oct 23rd 2011, 10:42 AM
Veve
Re: free group
if I choose the group with 2 generators to be ZxZ, is it solvable?
• Oct 23rd 2011, 11:05 AM
Deveno
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?
• Oct 23rd 2011, 11:15 AM
Veve
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...
• Oct 23rd 2011, 11:28 AM
Deveno
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)?