Imaginary Numbers under Addition form Group

Theorem

Let $\II$ denote the set of complex numbers of the form $0 + i y$

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

Then the algebraic structure $\struct {\II, +}$ is a group.

Proof

We have that $\II$ is a non-empty subset of the complex numbers $\C$.

Indeed, for example:

$0 + 0 i \in \II$

Now, let $0 + i x, 0 + i y \in \II$.

Then we have:

\(\ds \paren {0 + i x} + \paren {-\paren {0 + i y} }\)

\(=\)

\(\ds \paren {0 + i x} - \paren {0 + i y}\)

Inverse for Complex Addition

\(\ds \)

\(=\)

\(\ds i \paren {x - y}\)

Definition of Complex Subtraction

\(\ds \)

\(\in\)

\(\ds \II\)

Hence the result by the One-Step Subgroup Test.

$\blacksquare$

Sources

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