Is there any intuitive understanding of normal subgroup?

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 explanation or an example of this concept? Why it is very important in algebra?

• This is the way I think of things: A normal subgroup of an arbitrary group is the "correct" generalization of an arbitrary subgroup of an abelian group, with respect to factoring (taking quotients). Commented Nov 10, 2014 at 7:36
• This may be a duplicate of Intuition behind normal subgroups. Or perhaps not, since this question seems to be aimed at a lower level than that one. Commented Nov 10, 2014 at 7:37
• If you view subgroups of a group acting on some space as (possible) stabilizers, normal subgroups are those which stabilize not only points, but whole orbits. In that sense, they are global versions of subgroups. Commented Nov 10, 2014 at 19:15
• Commented Nov 11, 2014 at 8:18
• Commented Jun 8, 2022 at 4:34

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.

UPDATE: example of a normal subgroup.

Let's take the same cube. Now lets allow only rotations for 180 degrees around $$x$$, $$y$$ and $$z$$ axis. And any combination of such rotations. This will be group $$B$$. $$B$$ is a subgroup of the group of all rotations of cube. It is a proper subgroup of the original group (each face of our cube either remains in place or is moved to the opposite position by our rotations, so not all the elements of original group are included into this subgroup).

The definition of $$B$$ "does not depend of frame of reference", so I am sure this is a normal subgroup. Well, I understand that "does not depend of frame of reference" is not an accurate description, but this whole question is about intuition.

By the way, looks like group $$B$$ consists of only 4 elements: after any combination of described rotations only 1 of 2 faces can become the front, and only 1 of 2 faces can become the top. So "and any combination of such rotations" in my definition of group $$B$$ can be substituted with "and identity element $$e$$".

• Nice but unsufficient : for a proper grasp of the idea, a second example and a NONE example would be nice. Commented Aug 10, 2015 at 12:46
• Interesting. Does this extend to infinite-dimensional spaces, like let's say a change of numeraire in a time series? Commented Sep 20, 2015 at 21:46

A congruence in a group $$G$$ is an equivalence relation $$\equiv$$ in $$G$$ that is compatible with the operation of $$G$$: $$a \equiv b, \ a' \equiv b' \implies aa '\equiv bb'$$ The quotient $$\overline G = G\,/\equiv$$ is then a group.

It is easy to prove that, when $$\equiv$$ is a congruence in $$G$$, the equivalence class of $$1$$ is a normal subgroup $$N$$ of $$G$$ and the equivalence classes are the cosets of $$N$$.

Conversely, if $$N$$ is a normal subgroup of $$G$$, then the relation defined by $$a \equiv b$$ if $$a^{-1}b \in N$$ is a congruence relation in $$G$$ whose equivalence classes are the cosets of $$N$$.

So, normal subgroups are natural objects when you consider congruences. The other equivalent characterizations of normality, such as $$aN=Na$$ and invariance under conjugation, follow easily.

I learned this approach in Basic Algebra I by Jacobson. The same approach leads to ideals in ring theory.

Quotients of groups are only well defined if we take the quotient over a normal subgroup. Another way of writing your definition is that $N$ is normal iff $gN = Ng$, so the set of left cosets equals the set of right cosets, which makes the quotient group $G/N$ well-defined. That is, in my opinion the key reason to be interested in them. Otherwise, normal subgroups keep popping up literally everywhere in group theory. For example, when we look at solvable groups (Galois theory), they are important.

• Every subgroup's cosets partition the group. So the quotient group, or the set of cosets, can be defined as a group in many different ways. However, the composition rule for cosets is well defined with respect to representative elements iff the subgroup is normal, which, to clarify, is what I think Johanna means by G/N being well defined. Commented Jun 27, 2016 at 2:57