Rational Numbers under Multiplication do not form Group

Theorem
The algebraic structure $\struct {\Q, \times}$ consisting of the set of rational numbers $\Q$ under multiplication $\times$ is not a group.

Proof
that $\struct {\Q, \times}$ is a group.

By the definition of the number $0 \in \Q$:
 * $\forall x \in \Q: x \times 0 = 0 = 0 \times x$

Thus $0$ is a zero in the abstract algebraic sense.

From Group with Zero Element is Trivial, $\struct {\Q, \times}$ is the trivial group.

But $\Q$ contains other elements besides $0$.

From this contradiction it follows that $\struct {\Q, \times}$ is not a group.

Also see

 * Non-Zero Rational Numbers under Multiplication form Infinite Abelian Group