Consider addition modulo N relation $a \pmod n$ over all the integers for example. We can easily prove that this is a group. Identity is any element which satisfies the following:
$$a \cdot e=a$$
In the given relation $a \pmod n$, we can have $0$ as identity. But $n, 2n$ and so on. These numbers will also satisfy the above condition.
Similarly we can have multiple inverses by just adding $n, 2n$ and so on..
What am I missing here?