Show that $S=\mathbb{R}\setminus \left \{-1 \right \}$ is abelian.

Question

Can someone please give me a hint on how to solve this?

Let $S=\mathbb{R}\setminus \left \{ -1 \right \}$ and define a binary operation on $S$ by $a\circ b=a+b+ab$. Prove that $(S,\circ)$ is an abelian group.

I cannot just say $a\circ b=a+b+ab=b+a+ba=b\circ a$, so what should I do?

Answer 1

You must verify all of the abelian group axioms. You have shown that this binary operation is commutative, but it remains to show:

• This binary operation takes pairs of elements in the set, to another element in the set,
• This is an associative binary operation,
• An identity exists in the set,
• An inverse element exists in the set.

One is certainly allowed to utilise information about the field $\Bbb R$ to generate this proof.