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:

Hence the result by the One-Step Subgroup Test.