1. ## Structure theorems

If $N \triangleleft G$ and $G/N \cong (\mathbb{Z}, +)$, show that whenever $H,K \leq G$ with $H \cap K = N$ , then either H = N or K = N

Please post a complete proof to this, so I can learn from it. Its a study question for my exam. I really appreciate it.
Thanks guys

2. Originally Posted by ElieWiesel
If $N \triangleright G$ and $G/N \cong (\mathbb{Z}, +)$, show that whenever $H,K \leq G$ with $H \cap K = N$ , then either H = N or K = N

Please post a complete proof to this, so I can learn from it. Its a study question for my exam. I really appreciate it.
Thanks guys

What you're asked to do is equivalent to show that if $H\,,\,K\le \mathbb{Z}$ and $H\cap K=\{0\}$ , then $H=\{0\}\,\,\,or\,\,\,K=\{0\}$

Tonio

3. Hmmmm....I dont really see that Tonio. Can you be more specific....I would appreciate it.

Thanks

4. Originally Posted by ElieWiesel
Hmmmm....I dont really see that Tonio. Can you be more specific....I would appreciate it.

Thanks

As $G\slash N\cong \mathbb{Z}$ and $N=H\cap K\le H\,,\,K$ , we get two sbgps $H\slash N\,,\,K\slash N \le G\slash N\cong \mathbb{Z}$.

Now, saying that $H\cap K=N$ is the same as saying $H\slash N\cap K\slash N=\{\overline{0}\}$ , with $\{\overline{0}\}$ the unit element of $G\slash N\cong \mathbb{Z}$...

Tonio