Imaginary Numbers under Multiplication do not form Group

Theorem
Let $\II$ denote the set of complex numbers of the form $0 + i y$ for $y \in \R_{\ne 0}$.

That is, let $\II$ be the set of all wholly imaginary non-zero numbers.

Then the algebraic structure $\struct {\II, \times}$ is not a group.

Proof
Let $0 + i x \in \II$.

We have:

So $\struct {\II, \times}$ is not closed.

Hence $\struct {\II, \times}$ is not a group.