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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 $G$.2012-06-04

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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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    @scaahu: The wreath product is, of course, an application of direct products (since it requires direct products for its definition). And the wreath product of $G$ by $H$ contains a copy of $G\times H$ (it contains a copy of every group that has a normal subgroup isomorphic to $G$ and quotient isomorphic to $H$). What you are asking is just way too broad.2012-06-04