Geometric meaning of normal in group theory?

Question

How should one think about normal subgroups intuitively? Is there any useful geometric intuition behind them?

For instance, I remember reading somewhere that normal subgroups are like bundles in some sense.

Answer 1

Perhaps the most enlightening characterisation of normality is that a subgroup $H$ is normal in $G$ precisely when there exists a homomorphism $\psi \colon G\to G'$ to some group $G'$ whose kernel is $H$. In other words, the normal subgroups are precisely those that occur as kernels of homomorphisms.

As for a geometric meaning of normality, that may be unlikely, depending on your view of groups. If you view groups purely algebraically, then there is really no intuition for what a group is, and thus none for normality either. If you like to think of groups as the groups of symmetry of some (abstract) geometric shape, whereby each group element is thought of as a symmetry; i.e., a transformation of the ambient space which keeps the shape invariant, then normality represents a collection of such transformations which occur as the kernel of some homomorphism to another group $G'$. That other group is the group of symmetric of some other shape, and the homomorphism $\psi$ can be though of as a mapping between the ambient spaces which maps the first shape to the second shape. The subgroup then being normal means that the symmetric in the subgroup are all maps by $\psi$ to the identity, namely to the transformation that does nothing. So a normal subgroup identifies certain symmetries of the shape which can (by applying $\psi$) be transferred to another space and another shape therein where they are killed.

Answer 2

So I am not sure which bundle interpretation you are referring to, so I will try to exploit this terminology in hope to find what you were looking for.

First try: As soon as you have a non-trivial map $G\to H$ which is like a bundle, i.e. all fibers are non-trivial, you have a normal subgroup, namely the fiber at $0$. Note that such bundles are up to isomorphism of $H$ canonically in 1-1 correspondence to normal subgroups.

Second try: as soon as we are given such a bundle we can apply $K(-,1)$. But those are quite complicated spaces in general (at least for intuition).

Third try: This one is maybe the most geometrical and convenient. You can choose your favorite space $X$ which realizes $G$ as fundamental group. For a bundle $G\to G/N$ we can construct the corresponding cover. This will have the normal subgroup $N$ as fundamental group. But here you can see the maps and actions nicely. The keywords are the long exact sequence in homotopy, the isomorphism of the Deckgroup and $G/H$ and the geometric and algebraic interpretation of the action of $G/N$ on $N$. Note that to really look into this you have to choose a basepoint in the cover.