Is the Fundamental group of the circle abelian?

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SUMMARY

The fundamental group of the circle (S1) is definitively abelian, as it is isomorphic to the additive group of integers (ℤ), which is known to be abelian. The isomorphism preserves the abelian property, meaning that if one group is abelian, the other must be as well. This conclusion is supported by the fact that any isomorphism between groups maintains the structure of the groups involved.

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  • Basic concepts of algebraic structures, particularly the additive group of integers
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fraggle
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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
 
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The fundamental group of the circle is the integers, which is abelian.
 
fraggle said:

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
 

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