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Math Help - Finding subgroups of index 2 of the free group.

  1. #1
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    Finding subgroups of index 2 of the free group.

    I want to show the free group of two generators has exactly three subgroups of index 2. I was told to consider homomorphisms from F_{2} \rightarrow \mathbb{Z}

    I know F_{2} is the same as going round a wedge union of two circles - I take the paths around the circles to be a and b. I was considering a map a to 0, b to 1, then we have a homomorphism from F_{2} \rightarrow \mathbb{Z} (right?), and from the first isomorphism theorem, the kernel of this homomorphism should give a normal subgroup of F_{2} - so we get a subgroup with generators a, b^{2}.

    But the idea of this question is to find double coverings of the wedge union, which ab^{2} clearly isn't.

    Even if I could find subgroups index 2, I don't see how I would know when I've found all of them.

    My group theory is so bad, I would appreciate any help on this.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Finding subgroups of index 2 of the free group.

    Quote Originally Posted by ILikeDaisies View Post
    I want to show the free group of two generators has exactly three subgroups of index 2. I was told to consider homomorphisms from F_{2} \rightarrow \mathbb{Z}

    I know F_{2} is the same as going round a wedge union of two circles - I take the paths around the circles to be a and b. I was considering a map a to 0, b to 1, then we have a homomorphism from F_{2} \rightarrow \mathbb{Z} (right?), and from the first isomorphism theorem, the kernel of this homomorphism should give a normal subgroup of F_{2} - so we get a subgroup with generators a, b^{2}.


    But the idea of this question is to find double coverings of the wedge union, which ab^{2} clearly isn't.

    Even if I could find subgroups index 2, I don't see how I would know when I've found all of them.

    My group theory is so bad, I would appreciate any help on this.
    Eh?
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  3. #3
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    Re: Finding subgroups of index 2 of the free group.

    here, we have a happy accident: any subgroup of index 2 is automatically normal. so it's the kernel of a surjection F_2 \to \mathbb{Z}_2.

    well, that makes it easy: what are our choices for possible images of the generators a and b?

    EDIT: your first instincts weren't that bad: look at the pictures on page 3 here:

    http://www.math.oregonstate.edu/~mat...ngs2006/NS.pdf
    Last edited by Deveno; November 6th 2011 at 04:13 PM.
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