Result: Fix $a \in \mathbb{R}$. Then $(\mathbb{R} \backslash \{a\}, *)$ is a group, where our group operation is defined by $x*y = (x-a)(y-a) + a$.
One consequence of this is the standard fact that the set of nonzero real numbers forms a group under regular multiplication. (Just pick $a = 0$.)
As another, slightly more interesting option, we could let $a = -1$. Then we have the group $(\mathbb{R} \backslash \{-1\}, *)$, where the group operation is defined by $x*y = x + y + xy$.
My question: Has anyone seen this result (or a generalization of it) elsewhere?
It may already exist as an exercise in an Abstract Algebra text, but I suspect this particular type of group is really a specific instance of something far more general. Any references or remarks on its generalization would be most welcomed.