TITLE: Is there any intuitive understanding of normal subgroup?
QUESTION [25 upvotes]: As the define goes:

A subgroup $N$ of a group $G$ is called a normal subgroup if it is invariant under conjugation; that is, for each element $n$ in $N$ and each $g$ in $G$, the element $gng^{−1}$ is still in $N$.

My Question is: Can anyone give me an intuitive explaination or an example of this concept? Why it is very important in algebra?

REPLY [30 votes]: Let's take a look at the group of rotations of cube. It has a subgroup of rotations around vertical axis. This subgroup (let's call it $A1$) has 4 elements: rotate the cube for 0, 90, 180 or 270 degrees.
There is another subgroup: rotation around one of horizontal axes. Let's call it $A2$.
Subgroups $A1$ and $A2$ are obviously different. But still they look so very much alike! If there was someone else looking at our cube from different angle he could even fail to understand my descriptions of $A1$ and $A2$ "correctly" and confuse $A1$ with $A2$.
This is because $A1$ and $A2$ are conjugated. The $g x g^{−1}$ actually means "look at $x$ from another point of view", and $g$ defines this "point of view".
Subgroup is normal if it is very "symmetric". No matter from which point you look at the whole group $G$ the subgroup $N$ remains at place.