A basic group question

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.

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 $\{0\}$?

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It's called the trivial group. – anon Aug 6 '12 at 22:59
How can you argue with someone over whether something is a group? There is a small finite list of axioms to check. Is the operation associative? Does it have an identity? Does it have inverses? – Qiaochu Yuan Aug 6 '12 at 23:01
@qiaochu yuan Exactly! – Amihai Zivan Aug 6 '12 at 23:05
I'd like to ask, what is the argument against it? – Cocopuffs Aug 6 '12 at 23:12
It is too small a thing to argue about. But you are right. – André Nicolas Aug 6 '12 at 23:18

1. Closure $0\cdot0=0 \in G$
2.Associativity $(0\cdot0)\cdot 0 = 0(\cdot0\cdot0)$
3.Identity element is $0$.
4.Inverse element holds, because if not, eist $x\neq0 \in G$ that don't have Inverse element. Absurd.