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Math Help - Free groups and subgroups

  1. #1
    Senior Member I-Think's Avatar
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    Free groups and subgroups

    Let F_2 be the free group on two generators F_2=<x,y>. 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=<x,y,z>. 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?
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  2. #2
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    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.
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  3. #3
    Senior Member I-Think's Avatar
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    Re: Free groups and subgroups

    So considering the subgroup H=<x^2,xy,y^2>, and F_3=<a,b,c>,
    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?
    Last edited by I-Think; January 13th 2013 at 05:51 PM.
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  4. #4
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    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?
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  5. #5
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    Re: Free groups and subgroups

    find three letters in an alphabet that only has two.




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