# Isomorphisms of Product Groups - How to Prove?

• Mar 4th 2011, 02:29 PM
rtplol
Isomorphisms of Product Groups - How to Prove?
Hi MHF,

This is my first post and it's totally cool to ignore this as I have not contributed in helping anyone else yet, but I'm really in a spot and have no one to ask, and do plan on helping others so help would greatly be appreciated...

The Question I'm having trouble with is:
"Determine whether G is Isomorphic to HxK where $G = R^x , H = {+- 1}, K = +R$

That is, the group of reals defined multiplicatively, the group of plus/minus 1 and the group of reals that are positive

In all honesty, I just have no idea where to start. I have no idea how to prove an isomorphism without a mapping explicitly stated. I'm not looking for the answer, just a nudge in the right direction on how to find the mapping (I literally have no idea and google is very difficult with abstract algebra topics)

thanks so much.
• Mar 4th 2011, 02:56 PM
tonio
Quote:

Originally Posted by rtplol
Hi MHF,

This is my first post and it's totally cool to ignore this as I have not contributed in helping anyone else yet, but I'm really in a spot and have no one to ask, and do plan on helping others so help would greatly be appreciated...

The Question I'm having trouble with is:
"Determine whether G is Isomorphic to HxK where $G = R^x , H = {+- 1}, K = +R$

That is, the group of reals defined multiplicatively, the group of plus/minus 1 and the group of reals that are positive

So the question seems to be whether $\mathbb{R}^*\cong \{\pm 1\}\times \mathbb{R}_+$

Let's try $f:\mathbb{R}^*\rightarrow \{\pk 1\} \times \mathbb{R}_+\,,\,\,f(r):=(syg(r),|r|)\,,\,\,with\, \,syg(r)=\left\{\begin{array}{rr}1&\mbox{ , if } r>0\\-1&\mbox{ , if }r<0\end{array}\right.$

Check the above is a group isomorphism.

Tonio

In all honesty, I just have no idea where to start. I have no idea how to prove an isomorphism without a mapping explicitly stated. I'm not looking for the answer, just a nudge in the right direction on how to find the mapping (I literally have no idea and google is very difficult with abstract algebra topics)

thanks so much.

.
• Mar 4th 2011, 03:01 PM
rtplol
Thank you tonio!! That's literally all I needed to get started on a whole bunch of these questions.