Showing that $\mathbb{C}^\times$ is an abelian group

Question

QUESTION

Multiplication of complex numbers defines a binary operation on $\mathbb{C}^\times := \mathbb{C} \setminus \{0\}$. Show that $\mathbb{C}^\times$ together with this operation is an abelian group.

ATTEMPT

I know that for an abelian it has to show commutativity and for it to be a group there must be associativity of multiplication, but not sure where to go with it.

Do you have to show that it is a group, or do you know that it is a group and have to show that it is abelian? — Travis Willse, Jan 28 '16 at 14:15
I have to try to show that it is abelian, and yeah we know that it is a group — user281831, Jan 28 '16 at 14:26

score 4 · Answer 1 · answered Jan 28 '16 at 14:41

4

Big Hint

using the fact that $\mathbb R$ is abelian, we can very easily prove that

$$(a+ib)(c+id)=...=(c+id)(a+ib)$$ for $a,b,c,d\in\mathbb R$.

answered Jan 28 '16 at 14:41

Surb

55,662

Showing that $\mathbb{C}^\times$ is an abelian group

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