# Thread: Isomorphisms of Product Groups - How to Prove?

1. ## 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 $\displaystyle G = R^x , H = {+- 1}, K = +R$

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.

2. 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 $\displaystyle G = R^x , H = {+- 1}, K = +R$

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 $\displaystyle \mathbb{R}^*\cong \{\pm 1\}\times \mathbb{R}_+$

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

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.
.

3. Thank you tonio!! That's literally all I needed to get started on a whole bunch of these questions.