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Homework Help: Is the Fundamental group of the circle abelian?

  1. Nov 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Is the Fundamental group of the circle (S^1) abelian?

    Not a homework question, just something I want to use.

    2. Relevant equations

    3. The attempt at a solution
    Intuitively it appears to be and it is isomorphic to the additive group of integers which is abelian. I believe isomorphism preserve the abelian property. I'd just like a second opinion.

  2. jcsd
  3. Nov 15, 2009 #2
    The fundamental group of the circle is the integers, which is abelian.
  4. Nov 15, 2009 #3


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    Staff Emeritus
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    If you have an isomorphism from G to H, call it f, and H is abelian then

    f(gh)=f(g)f(h)=f(h)f(g) (since H is abelian) = f(hg)

    So f(gh)=f(hg) which means gh=hg since f is a bijection. Hence G is abelian too
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