Let $G=\{0, \cdot\}$.
I'm arguing with someone over if $G$ is a group with the regular multiplication since I don't see why it isn't.
Addition:
Now, $G=\{\mathbb{Z},\triangle \}$ with $x \triangle y=x+y+xy$. Is it true that $G$ is not a group and the only subset of $\mathbb{Z}$ to form a group with $\triangle$ is $\{0\}$?