Group Theory. Isomorphic Groups on the Set R. G is the set of the real numbers with the operation x*y=x+y+1. Find an isomorphism f: R-G and show that it is an isomorphism.

Question

Group Theory.  Isomorphic Groups on the Set R.  G is the set of the real numbers with the operation x*y=x+y+1.  Find an isomorphism f: R->G and show that it is an isomorphism.

kinggeorge · Answer

Do you know what the identity of G is?

anonymous · Answer

No

anonymous · Answer

Well, maybe....-(x+y+1)

kinggeorge · Answer

In fact, the identity, would be -1. You can solve for it by looking at the equation $xy=x+y+1=y$. Subtract $y$, and you get $x+1=0\implies x=-1$. Since any isomorphism maps identity to identity, you know your map sends $0\mapsto-1$. 

This should suggest an isomorphism. In particular,$$\begin{aligned}f:R&\longrightarrow G\x&\longmapsto x-1\end{aligned}$$

kinggeorge · Answer

I think this map should work. You just need to prove it's an isomorphism.

anonymous · Answer

Thank you for your help!  i've struggled with this problem all weekend.

kinggeorge · Answer

No problem.