Importance of Normal subgroups

Question

Why we study the normal subgroups? What is the importance of normal subgroups in group theory? My teacher ask today that how you think that normal subgroups are important. I told that it is necessary for the factor groups but he was not satisfied with this answer.please tell me the importance of normal subgroups. Thanks

Answer 1

Normal subgroups are important because they are exactly the kernels of homomorphisms.

In this sense, they are useful for looking at simplified versions of the group, via quotient groups.

Answer 2

One way you can answer for this question is normal subgroups allow you to define quotient group which usually has simpler structure than the whole group. So this is like a shadow of a group $G$ on $G/N$ for normal subgroup $N$. So if you have many normal subgroups then you can take the lots of shadows and by looking at these shadows of $G$, you can deduce some information about group $G$.