Integers under Multiplication do not form Group

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

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

From Integers under Multiplication form Monoid, $\struct {\Z, \times}$ forms a monoid.

Therefore, and  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 $\struct {\Z, \times}$ is not a group.