# Prove direct product of G1 x G2 is abelian

1. Oct 31, 2010

### kathrynag

1. The problem statement, all variables and given/known data

Prove that if G1 and G2 are abelian, then the direct product G1 x G2 is abelian.

2. Relevant equations

3. The attempt at a solution
let G1 and G2 be abelian. Then for a1,a2,b1,b2, we have a1b1=b1a1 and a2b2=b2a2.
The direct product is the set of all ordered pairs (x1,x2) such that x1 is in G1 and x2 is in G2.
Let x1 be in G1 and x2 be in G2.
Then G1 x G= (x1, x2)
I'm not quite sure how to show this is abelian.

2. Oct 31, 2010

### micromass

Take two elements in G1 x G2. What does these elements look like? What happens if you multiply them?

3. Oct 31, 2010

### kathrynag

Let a be in G1 b in G2. Then ab=ba

4. Oct 31, 2010

### micromass

No, I take an element in G_1 x G_2. What does this look like?

5. Oct 31, 2010

### kathrynag

oh, so we have (a, b) and (c,d)
We have (a,b)(c,d)=(ac,bd)
But ac=ca and bd=db
So (ac,bd)=(ca,db)=(c,d)(a,b)

6. Oct 31, 2010

### micromass

Yes! good job!

7. Oct 31, 2010

### kathrynag

That's really all I have to do? it seems so simple.

8. Oct 31, 2010

### micromass

Yes, it's that simple