How should normal subgroups be introduced?

Question

One standard definition of a normal subgroup is

A subgroup $N \subset G$ is normal iff the set of left cosets $\{gN\}$ and right cosets $\{Ng\}$ coincide.

There's a class of similar definitions (every left coset is also a right coset, $gNg^{-1}=N$, etc). This definition doesn't seem to provide intuition to students about why we care, nor is it immediately obvious that this is necessary to form a quotient group. One soon learns, though, that

A subgroup is normal iff it is the kernel of some homomorphism.

This latter statement answers both of the previous complaints; it shows both why we care and the necessity of normality to take quotients. The first definition is more malleable for proving things, and the latter provides better intuition. Hopefully both are emphasized in an introductory course on group theory, but which should be taught first?

Answer 1

First of all, I should point out that the standard definition of a normal subgroup is

A subgroup $N \subset G$ is normal iff $g n g^{-1} \in N$ for all $n\in N$ and $g\in G$.

When I say "the" standard definition, I mean that this is how working group theorists think of normal subgroups, and this is one of two basic ways to prove that a subgroup is normal. (Note that this is a bit easier to use than the statement $N = gNg^{-1}$, since we don't have to prove containment in both directions.)

Of course, there is also the statement

A subgroup is normal iff it is the kernel of some homomorphism.

This is widely regarded as a theorem, as opposed to a definition, although it is certainly both the motivation for the definition of normal subgroups and the explanation for their importance.

The statement

A subgroup $N\subset G$ is normal iff $gN = Ng$ for all $g\in G$

appears as a definition in some books (Gallian being the chief culprit I am aware of), but for the life of me I can't figure out why. This is not a very good definition for proving things, nor for developing intuition about normal subgroups, nor for explaining their importance, nor is it very standard, and frankly it's not even one of the most important properties of normal subgroups.

When I teach this subject in abstract algebra, I start by introducing quotient groups, which I describe as a smaller group obtained from a partitioning a larger group into subsets. Next, I observe that you need to be careful about which subsets you choose if you want to get a well-defined group operation. This motivates the definition of a congruence relation, and I show that the equivalence classes for a congruence relation form a group. This mirrors the approach that is usually taken for defining modular arithmetic, and so immediately connects the two in the students' minds. It also has the advantage that the definition seems fairly natural.

I then go on to observe that the congruence class of the identity must be a subgroup that is closed under conjugation, which prompts the definition of a normal subgroup. I also point out to the students that congruence relations aren't used very often in group theory, since all you really need to know for making a quotient is the normal subgroup (although group theorists often use the notation $gN = hN$ when they want to say that $g$ and $h$ are congruent). I do homomorphisms a bit later on.

If I were prohibited from using this approach, I think I would start by introducing homomorphisms, and then observe that the kernel of a homomorphism is a subgroup that is closed under conjugation, prompting the definition of a normal subgroup. I would not define a normal subgroup as the kernel of a homomorphism, since that's really more of a theorem than a definition, but I would use homomorphisms to motivate the definition.

Answer 2

The way I like to approach this is as follows. After discussing subgroups, the natural question as to forming the quotient $G/H$ arises. I then proceed to look at the cosets and prove that if $gH\cdot g'H=gg'H$ is a well-defined operation, then the cosets become a group, which we call the quotient group. This is a very easy proof with basically nothing to do. So, one clearly sees that the obstruction to forming the quotient group is the well-defindedness of the operation above. And voiala, this operation is well-defined if, and only if, $gH=Hg$ (or any of the other equivalent conditions). I find that this motivates the definition of normality clearly. I then go over the standard equivalent definition, emphasizing what each is good for (i.e., for checking normality of a given subgroup use that condition, for proving general things about normal subgroups use that etc.).

Answer 3

I like Gowers' fake history of normal subgroups.

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This is also good. Especially if you can relate it to change-of-basis, and Weyl's famous quotation "The introduction of a coördinate system is an act of violence".

Some resources I like for the basis-free approach:

• http://linear.axler.net
• https://unapologetic.wordpress.com
• https://sites.google.com/site/winitzki/linalg
• http://faculty.luther.edu/~macdonal/laga/

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For an application to finance, note that "change of basis" = "change of numeraire". Aaron Brown calls change-of-numeraire "the most important tool" or something like that. (Also note in Emanuel Derman's advice to quants that saying complete markets exist because of Girsanov's theorem is considered bad form (putting the model above reality).

Every currency, or perhaps any bond or share class, can be considered as a "numeraire" (every publicly-traded corporation could be viewed as issuing its own currency. The ability to buy eg other firms partly by giving stock and not only having to use cash, is one of the principal reasons cited to go public.

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Change-of-basis also shows up famously in image compression so again there's a practical reason to look at change-of-basis. Quotients of kernels can then be sold as "deeper, simpler thinking" about change-of-basis, once it's been accepted as important.

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So this is perhaps a sketch for one long detour off the path to normal subgroups per se, onto the topic of change-of-basis. But if Gowers treats the small ($S_3$) case extremely well, and my analogy between change-of-basis and commutative subgroups is correct, then you can wave your hands in that direction with some pointers to further reading.

Answer 4

I spend some time talking about this in my question here

https://mathoverflow.net/questions/41955/does-any-textbook-take-this-approach-to-the-isomorphism-theorems

I really think defining normal subgroups as kernels of homomorphisms is a great idea.

You can also find a lot of support for introducing normal subgroups as kernels here: https://mathoverflow.net/a/7796/1106

Answer 5

Not being an algebraist, I am not annoyed with cheating a bit (well, we can use fictional history, as Tim Gowers defends so well...)

Start with a fundamental, natural question: given a polynomial $p(x)$ with integer coefficients, find a numerical set of sorts (i.e., a small field) with a root of $p$.

We all know the exquisite answer: consider 'arbitrary' polynomials and operate $\mod p$. Which in turn gets codified by passing a quotient by... an ideal (and irreducibility comes up...).

A normal subgroup is an ideal's smaller brother, in a rather tight context, when only one well behaved operation is being considered.