1. ## Exact sequence 'counterexample'

I would like to show that if the sequence:

$0 \to \mathbb{Z} \to G \to \mathbb{Z}/2\mathbb{Z} \to 0$

is exact, then G need not necessarily be isomorphic to $(\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z}$. Just one example of G that is not isomorphic to this (where the above sequence is still exact) would be enough. I've tried many and I can't seem to find a single valid one.. any hints?

2. This is not my area of expertise, so excuse me if I make a dumb error, but can't we just let G be the integers with the first map multiplication by 2, and the second map sending n to n(mod 2)?

Edit: I didn't check if $\mathbb{Z}$ is isomorphic to the direct sum. The obvious map is probably an isomorphism.

3. If I'm not mistaken, you can take $G = \mathbb{Z}$ and define $\Phi : \mathbb{Z} \to G$, $\Psi : G \to \mathbb{Z}_2$ by
$\Phi (1) = 2, \ \Psi(k) = k (mod 2)$

then $\Phi$ is injective, $\Psi$ is surjective and $Im \Phi = Ker \Psi$, so the sequence is exact.

Edit: Ah, Steve beat me to it. Woops.

4. Originally Posted by DrSteve
This is not my area of expertise, so excuse me if I make a dumb error, but can't we just let G be the integers with the first map multiplication by 2, and the second map sending n to n(mod 2)?

Edit: I didn't check if $\mathbb{Z}$ is isomorphic to the direct sum. The obvious map is probably an isomorphism.
Originally Posted by Defunkt
If I'm not mistaken, you can take $G = \mathbb{Z}$ and define $\Phi : \mathbb{Z} \to G$, $\Psi : G \to \mathbb{Z}_2$ by
$\Phi (1) = 2, \ \Psi(k) = k (mod 2)$

then $\Phi$ is injective, $\Psi$ is surjective and $Im \Phi = Ker \Psi$, so the sequence is exact.

Edit: Ah, Steve beat me to it. Woops.
Yes, that is exactly the problem! Virtually any decent map you can come up with gives $G=\mathbb{Z}$ and this is isomorphic to $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}$. So really you need to be able to find something quite strange, like an exact sequence where $G =\mathbb{Z}^2$. But I've tried many functions and it seems all but impossible.

5. Wait a second, $\mathbb{Z}$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}$. The latter group has torsion elements, like $(1,0)$. The former group is torsion-free. (You could also invoke the Fundamental Theorem of Finitely Generated Abelian Groups.)

6. Originally Posted by roninpro
Wait a second, $\mathbb{Z}$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}$. The latter group has torsion elements, like $(1,0)$. The former group is torsion-free. (You could also invoke the Fundamental Theorem of Finitely Generated Abelian Groups.)
Great! Don't know how I missed that.. the 'obvious' homomorphism between them looked like an isomorphism to me.

7. Originally Posted by Capillarian
Great! Don't know how I missed that.. the 'obvious' homomorphism between them looked like an isomorphism to me.
You might need to check the difference between a short exact sequence and a split exact sequence. Being a short exact sequence is a necessary condition for being a split exact sequence. Given a short exact sequence, there are some additional conditions to be satisfied for being a split exact sequence.