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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?

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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

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Normal subgroups are important for the same reason that factor groups are important, since normal subgroups correspond to factor groups and vice-versa.

Direct products (whether internal or external; they correspond to one another in a natural way) give you both ways of producing new groups from old, and of (sometimes) understanding more complicated groups in terms of simpler ones. A classical example of the latter is the Fundamental Theorem of Finitely Generated Abelian Groups (which is later generalized to any finitely generated module over a PID), which tells you that any finitely generated abelian group is a direct product of cyclic groups that, in addition, have orders satisfying certain restricting relations. This makes understanding finitely generated abelian groups very easy.

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Arturo, what about direct products of non-abelian groups, how are they important in the sense that the direct product produces more interesting properties? –  scaaahu Jun 4 '12 at 4:11
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@scaaahu: I don't undestand what you mean by "produces more interesting properties". Direct products of groups can help you understand more complicated groups in terms of simpler ones, whether the simpler ones are abelian or not. There are many theorems associated to direct products and subdirect products, and direct product play an important role in constructions, particularly limits (in the sense of category theory). –  Arturo Magidin Jun 4 '12 at 4:19
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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
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@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
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Thanks, @Arturo. That's what I was looking for. Do you know a good book/paper which dicuss this issue in details, i.e. direct product of non-abelian groups? Many algebra books discuss direct product/decomposition of abelian groups. I cannot seem to find a good one talking about non-abelian groups. Many books all of a sudden turn into wreath products. Or I just missed them because I am an absent-minded reader. –  scaaahu Jun 4 '12 at 5:09

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