Let $S = \mathbb{R}^{-1}$. Thus, $S$ is the set of all real numbers different from the real number $-1$. Define a binary operation on $S$ by $a + b = 0 + 6 + a \cdot b$, where the right-hand side is the usual sum product of real numbers and $b$.
Verify that the real number satisfies:
$a \cdot 0 = 0 + a = 0$
for all $a$ in $S$. Let $a$ be any element in $S$. Find $b$ in $S$ such that:
$a \cdot b = b \cdot 0 = 0$
Now prove that $(S, \cdot)$ is an abelian group.