# Normal Subgroups and Internal Direct Products

Why are Normal Subgroups important?

Why are Internal Direct Products important?

I'm studying abstract algebra and I have always wondered about its relevance and usefulness. Does anyone could help me please?

-
For starters, you can’t seriously study groups without simultaneously studying group homomorphisms, and the moment you do that, you’re looking at normal subgroups: a subgroup $N$ of $G$ is normal iff it’s the kernel of some homomorphism with domain $G$. – Brian M. Scott Jun 4 '12 at 3:32

I was thinking the other way around. I have two non-abelian groups, $G$ and $H$. What are the properties that $G \times H$ has but $G$ or $H$ don't? And could we have the case that $G$ and $H$ have some properties but $G \times H$ doesn't share? – scaaahu Jun 4 '12 at 4:34
@Scaahu: That's too broad a question; for example, if $G$ and $H$ are simple nontrivial, $G\times H$ is not simple. $G\times H$ has proper normal subgroup, neither $G$ nor $H$ do. And so on. – Arturo Magidin Jun 4 '12 at 4:36