Why is $\mathbb{Q}[i]$ not a subgroup of $\mathbb{R}$?

Question

$i$ is the imaginary unit on the complex plane.

I am confused on a particular portion of the definition of a subgroup.

Let $G$ be a group and $H$ is a subset of $G$.

A part of the definition is stated as follows: "The restriction of the group operation to $H$ takes values in $H$"

Let $+: G \times G \to G$

So I am interpreting that portion of the definition as $+|_H: H \times H \to G$ or is the codomain supposed to $H$? That's what I'm not clear on...

And how $\mathbb{Q}[i]$ is not a subgroup of reals... Thanks! I just started to learn on group theory so details would be appreciated, otherwise i would not understand...

Also from the "one step test of subgroups" I don't think it fails?

Answer 1

$\mathbb{Q}(i)$ is not a subsets of the set of real numbers, so it is not a subgroup.