Integers under Multiplication do not form Group

Theorem
The set of integers under multiplication $\left({\Z, \times}\right)$ does not form a group.

Proof
In order to be classified as a group, the algebraic structure $\left({\Z, \times}\right)$ needs to fulfil the group axioms.

From Integers under Multiplication form Monoid, $\left({\Z, \times}\right)$ forms a monoid.

Therefore group axioms $G0$, $G1$ and $G2$ are satisfied.

However, from Invertible Integers under Multiplication, the only integers with inverses under multiplication are $1$ and $-1$.

As not all integers have inverses, it follows that $\left({\Z, \times}\right)$ is not a group.