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:

\(\ds \paren {0 + i x} \times \paren {0 + i x}\)

\(=\)

\(\ds \paren {0 - x^2} + i \paren {0 \times x + 0 \times x}\)

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds -x^2\)

\(\ds \)

\(\notin\)

\(\ds \II\)

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

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

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{B ii}$