# Structure theorems

• December 1st 2009, 09:21 AM
ElieWiesel
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
• December 1st 2009, 10:03 AM
tonio
Quote:

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
• December 1st 2009, 10:10 AM
ElieWiesel
Hmmmm....I dont really see that Tonio. Can you be more specific....I would appreciate it.

Thanks
• December 1st 2009, 10:58 AM
tonio
Quote:

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