# Products of groups with no non-abelian quotients

Suppose I have two groups $G$ and $H$ with no non-abelian quotients. Then does $G \times H$ have no non-abelian quotients?

-
Did you mean Abelian quotients? If $G$ and $H$ each have no non-trivial Abelian quotient group, then each is a perfect group, so $G \times H$ is a perfect group, and has no non-trivial Abelian quotient group –  Geoff Robinson Aug 27 '12 at 13:45

Every group is a quotient of itself, so if $G$ and $H$ have only abelian quotients then in particular $G$ and $H$ are abelian, and so is $G \times H$. Since every quotient of an abelian group is again abelian, $G \times H$ has only abelian quotients.
@Auke, see Jacob's remark, if $G$ and $H$ would be non-abelian simple groups, then $G \times H$ would have non-abelian quotients ... –  Nicky Hekster Aug 27 '12 at 14:38