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Math Help - abelian groups

  1. #1
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    Question abelian groups

    Is there a group that satisfies a^3=1 for each a in G but isn't abelian?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by MathBird View Post
    Is there a group that satisfies a^3=1 for each a in G but isn't abelian?
    I can't find any reason why not - but if it is finite then it is at least 3-generated (because it is regular so it can't be 2-generated).

    What about <x,y,z|x^3=y^3=z^3=1, xyx^{-1}=y^3, xzx^{-1}=z^3, yzy^{-1}=z>, the semidirect product of C_3 \times C_3 with C_3? I'm not sure if that is actually properly defined - I'm doing this in a bit of a hurry...
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  3. #3
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    Thanks a lot. But may I ask for a little more details.
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by MathBird View Post
    Thanks a lot. But may I ask for a little more details.
    About the 3-generated bit? I was making it too complicated, and I think my logic was wrong. However, the group must be at least 3-generated if it is finite as otherwise it will have order 3 or 9, and there are no non-abelian groups of order 3 or 9. Also,
    <x,y,z; x^3 = y^3 = z^3 = 1, yz = zyx, xy = yx, xz = zx>

    is a group that works. I believe the group I gave earlier is nonsense...

    EDIT: Of course it's nonsense - y and z are just the identity element...
    Last edited by Swlabr; September 9th 2009 at 08:22 AM.
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