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?
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.