Is a set $w \cdot z = w + z + iwz$ an abelian group?

Question

Let $w \cdot z = w + z + iwz$ for every $w,z \in \mathbb{C}$

1) Is $(\mathbb{C},\cdot)$ an abelian group?
2) Is $(\mathbb{C} \setminus \left\{ i\right\} ,\cdot)$ an abelian group?

Regarding 1) I checked that every condition is true, except inverse element because it doesn't exist for $x=i$, so $(\mathbb{C},\cdot)$ is not an abelian group.

I don't know how to to this for 2) though.

Answer 1

You have already covered associativity, commutativity, and identity in 1).

Restricting the set takes care of the problem you had with inverses in 1)

All that remains is closure:

In 1) you have shown that $w,z \in \mathbb C \implies w\cdot z \in \mathbb C$

But what about $w,z \in \mathbb C\setminus\{i\}$?

Suppose $\exists w,z \in \mathbb C\setminus\{i\}:w\cdot z = i$

$iwz + w + z - i = 0$

Which can be factored: $i(w-i)(z-i) = 0$

$w\cdot z = i \iff w=i,$ or $z = i$

But neither are in $C\setminus\{i\}$