Infinite cyclic groups isomorphic to Z

[radou]

I'm currently going through Hungerford's book "Algebra", and the first proof I found a bit confusing is the proof of the theorem which states that every infinite cyclic group is isomorphic to the group of integers (the other part of the theorem states that every finite cyclic group of order m is isomorphic to the group Zm, but I understood that part).

So, if G = <a> is some infinite cyclic group, then the mapping f : Z --> G with the formula f(k) = a^k is clearly an epimorphism. To prove that it's an isomorphism (i.e. that Z is isomorphic to G), one needs to show that f is a monomorphism, too. That's where I'm stuck. I know that f is injective iff Ker(f) = {0}, so I somehow need to show that Ker(f) = {0}.

So, by definition, Ker(f) = {k in Z | a^k = e}. Suppose Ker(f) is non-trivial. Ker(f) is a subgroup of the integers Z, and hence it is cyclic, infinite and generated with m, where m is the least positive integer in it, so Ker(f) = <m>.

Thanks in advance.

[morphism]

If a^k=e, what can you say about the order of a?

[radou]

If a^k=e, what can you say about the order of a?

Well, the next theorem states that, if a has infinite order, then a^k = e iff k = 0, and hence, everything is clear. But, there is no proof for this theorem, it merely says that it is "an immediate consequence of the former theorem (the one I'm trying to prove).

[morphism]

OK, how about this: if ker(f)=<m>, try to define an explicit isomorphism from Z/<m> onto G (use f).

I gotta run. Hopefully this'll be good enough!

[radou]

OK, how about this: if ker(f)=<m>, try to define an explicit isomorphism from Z/<m> onto G (use f).

I gotta run. Hopefully this'll be good enough!

Well, this is how the proof in the book goes on - it is shown that [k] --> a^k is an isomorphism between Zm and G, but this means that G is a finite cyclic group, doesn't it? I still don't inderstand how it's proved for infinite cyclic groups. Actually, the order of G isn't mentioned in any part of the proof. I seem to be missing something big here.

[morphism]

But that's exactly it - if the kernel of f isn't trivial, then G cannot be infinite.

[radou]

But that's exactly it - if the kernel of f isn't trivial, then G cannot be infinite.

Forgive my (probable) stupidity, but why does this hold? Intuitive, it is somehow clear to me, but then again, why couldn't a^k = e, for some non zero k hold and G still be infinite?

[morphism]

If the kernel of f is non-trivial, then:
it is shown that [k] --> a^k is an isomorphism between Zm and G, but this means that G is a finite cyclic group

So...

[n_bourbaki]

This is so utterly trivial that I cannot believe it needs a discussion.

G is infinite and cyclic and G=<a>, i.e. G is the set of all integer powers of a if written multiplicatively. So in what way is the map

a^k ---> k

not a clear isomorphism? All powers of a are distinct.

[radou]

If the kernel of f is non-trivial, then:


So...

I just realized that this is more a matter of logic than of math. Thanks. :smile: