Is the Fundamental group of the circle abelian?

[fraggle]

Homework Statement

Is the Fundamental group of the circle (S^1) abelian?

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

Homework Equations

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.

Thanks

 

[n!kofeyn]

The fundamental group of the circle is the integers, which is abelian.

 

[Office_Shredder]

Homework Statement

Is the Fundamental group of the circle (S^1) abelian?

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

Homework Equations

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.

Thanks

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