Prove that, if the direct product A1 x A2 is Abelian, then

[Laurenmath]

Let (A1, *) and (A2, **) be groups, using the defination of Abelian group. Prove that, if the direct product A1 x A2 is Abelian, then the group A2 is Abelian.

So far, I have:
(a1, a2) * (a1',a2') = (a1',a2') ** (a1,a2)

(a,b) = (c,d) if and only if a = c and b = d

 

[daon]

You defined A1, A2 as abelian groups and are asked to show that A2 is abelian?

Are you sure the question isn't: "Let A1, A2 be groups and A1xA2 abelian. Then A2 is abelian."

A1xA2 abelian means: Let (a1,a2),(b1,b2) be elements of A1xA2. Then (a1,a2)*(b1,b2) = (b1,b2)*(a1,a2), where * is the operation of the direct product (group operations done componentwise).

Then (a1,a2)*(b1,b2)=(a1*b1, a2**b2)

Also, (a1,a2)*(b1,b2)=(b1,b2)*(a1,a2) = (b1*a1, b2**a2).

By transitivity, (a1*b1,a2**b2) = (b1*a1,b2**a2).

-Daon