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?
 A: 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.
A: It is important because all those studies lead e.g. to Galois Theory. Galois Theory is important e.g. in solving polynomial equation. Solution to polynomial equation is of course important as it solves no one knows how many problems in real world from theoretical physics to engineering and economy.
I feel your tone though. So, YES, most of the teachings are bad in a sense that they:


*

*Ignore the intention of a theorem that we are left in the dark and missing the excitement;

*Skip too many critical arguments by assuming the students somehow could been suddenly skilled in tens of theorems just presented earlier;

*Give too many puzzles to be forgotten later that the sole purpose seems to just slow down the learning.
Self-learning is almost impossible as they don't seem to care the learning process of a student.
