1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Is the Fundamental group of the circle abelian?
  1. Abelian Groups (Replies: 5)

  2. Abelian Groups (Replies: 9)

  3. Abelian group (Replies: 2)