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Math Help - free group

  1. #1
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    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.
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  2. #2
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    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.
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  3. #3
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    Re: free group

    if I choose the group with 2 generators to be ZxZ, is it solvable?
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  4. #4
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    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?
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  5. #5
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    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...
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  6. #6
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    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)?
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