Suppose I have two groups $G$ and $H$ with no non-abelian quotients. Then does $G \times H$ have no non-abelian quotients?
Products of groups with no non-abelian quotients
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group-theory
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0Did 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 – 2012-08-27
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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.
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0@Auke, see Jacob's remark, if $G$ and $H$ would be non-abelian simple groups, then $G \times H$ *would* have non-abelian quotients ... – 2012-08-27