# Thread: Show a quotient group is abelian

1. ## Show a quotient group is abelian

Let A be an abelian group and let B be a subgroup of A. Prove that A/B is abelian.
So this is what I have:

Let $x, y \in A$, then $xB, yB \in A/B$. We must show $xB \cdot yB = yB \cdot xB$
Since B is a subgroup of A and A is abelian, then B is a normal subgroup of B.
We compute:
$xB \cdot yB = (xy)B$ because B is a normal subgroup of A.
$= (yx)B$ because A is abelian
$= yB \cdot xB$

And so A/B is abelian. Is this right?

2. ## Re: Show a quotient group is abelian

Yes, that is correct.