I am in the process of writing a longer post on Galois Theory (see here), and one of the central concepts is that of a normal subgroup.
We all know the definition (and their equivalents) from classes/books, but anyone who likes to ‘see their mathematics’ is left with the question:
… but what do they look like?
In this post I give a few different interpretations of the standard definitions, and go some way to explaining why it is difficult to answer this question in concrete terms.
Spoiling the punchline a bit, it seems to me that problem is actually the other way round: most of the groups that you have an intuition about are normal, so the question is really when is a group not normal?
The standard definitions
Nothing new yet:
A subgroup of a group is normal if any of the following conditions hold (so if one is true they will all be):
- (‘left cosets equal right cosets’),
- is the kernal of a homomorphism .
Some examples to bear in mind
- If is abelian then all its subgroups are normal.
- If is something like then the natural projections give us two normal subgroups: and
- The symmetric group has the Klein Vierergruppe as a normal subgroup:
- The group of symmetries of a square, thought of as a subgroup of is not a normal subgroup of .
Intuition 1: normal subgroups are like projections
One of the key results for normal subrgoups is that they can be used to define the quotient group, and the isomorphism theorem says that the quotient group effectively shows that you have a group structure within once you look at ‘modulo ‘.
At this point a picture is nice (I got the idea from a post in Rip’s Applied Mathematics Blog, here).
If is a subgroup (hold back on the ‘normal’ for a moment) then we can organize into fibres. To do this we start with and then begin multiplying by elements of which are not in : either we get a new fibre or we get one we have seen before, and eventually we’ll get all the fibres. We can label each fibre by the first that generated the fiber.
Picture of fibres
This picture suggests to me that the fibres are somehow orthogonal to the labels.
When the subgroup is actually a normal subgroup then we know that the fibres will have a group structure too.
Consequently, the group looks as though it can be decomposed as something like ‘‘.
For this reason the projection example is a good one to bear in mind.
What makes something not a normal subgroup?
The symmetries of a square can be seen as a subgroup of . Just because it is not difficult, I’ll explain how I think of this in terms of pieces of paper and cardboard:
Cut out a square from a piece of cardboard and write the numbers 1, 2, 3 and 4 on its corners. Draw a slightly-bigger square on a piece of paper and labels its corners with ‘positions’ 1, 2, 3 and 4 so that when you put the cardboard square onto the piece of paper all the same numbers sit next to each other. This is the starting point.
Any move you apply to the cardboard square will generate a permutation of the numbers — for example a quarter turn will send the corner labelled 1 into the position labelled 2, and so on.
You see how it works.
This gives us a subgroup, called the dihedral group :
I would have bet a little bit of money that this could have been a normal subgroup … but it is not, and now we think about why it isn’t.
The unilluminating answer is that it is not normal because:
It is a good point to introduce some new notation:
we define to be .
I like this notation, because it suggests a lot of things that are true.