On the direct product of abelian groups

Question

1. Given that $A$ and $B$ are abelian groups, prove that the direct product of them $$A\times B$$ is an abelian group.
2. Given that $A\times B$ is an abelian group, prove that $A$ and $B$ are both abelian.

I used the basic property of Descartes product: $$(a,b)\cdot(c,d)=(a\cdot c,b\cdot d)$$ to prove each argument.

1. $(a,b)(c,d)=(ac,bd)=(ca,db)=(c,d)(a,b)$;
2. $(a,b)(c,d)=(c,d)(a,b)\implies (ac,bd)=(ca,db)\implies ac=ca, bd=db$.

I am not sure whether this property of Descartes product can be applied to groups. This question appears to be the 30th question out of 36 questions in an exercise, so I guess it's not that easy.

This is my first post, so please forgive me for any wrongdoing!:)

Answer 1

Of course you can! And in fact you are using the definition of direct product of groups; there is nothing wrong here.