# How should normal subgroups be introduced?

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?

• Since I've never taught this subject, I'm speaking as a learning person and not as a teaching person. But I didn't really get normal subgroups until I saw them as the natural result of a homomorphism. More specifically, the approach in "Visual Group Theory" really worked for me in a way that my Artin-based undergrad course didn't. – Michael Pershan Mar 16 '14 at 13:16
• I think one has to be careful about ex post facto judgements of clarity. "Subgroup of $G$" itself could equally well be defined as "the isotropy set [we can't call it a subgroup yet!] of a point in some $G$-space", which I think is lovely, but surely not the appropriate first definition. – LSpice Sep 6 '14 at 7:01
• My philosophy is very traditional: Definitions first, then theorems, proofs, and applications. I think that trying to motivate the "why we care" theorems/applications up front muddles the presentation and confuses students. Do it quickly, fine, but get the definitions of terms on the board in the students' vision ASAP. Discovery-based investigations can quickly suck up class time and be confusing to students. – Daniel R. Collins Sep 21 '15 at 5:45
• I’d add to this question that it would be nice to have some mention, early, of how normal subgroups work in the continuous case. But how? ―― The more types of spaces one can introduce at first to say "Here’s how quotient groups show up here, here, and here", the easier to justify that normal subgroups really are the way to tie things together. – isomorphismes Feb 5 '16 at 6:16

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.

• (+1) I'm just not so sure that all group theorists think of normal group a la that definition. Granted, most group theorists don't really care about this or that definition, but there are probably two ways your standard group theorist would think of a normal group. One is element-centric, and then probably the definition you mention is the one of choice. Another is more universal-property centric, in which case a normal group is probably thought of as the kernel of a homomorphism. – Ittay Weiss Mar 16 '14 at 5:39
• @IttayWeiss, to introduce the notion I agree that an element-centric definition is easier to grasp. – vonbrand Mar 16 '14 at 11:37
• I disagree that "element" based definitions should come before universal constructions. Beginning algebra is a great place to start thinking about objects in terms of the properties that characterize them, so I think it is often appropriate to "stretch" students with the universal definitions. – Steven Gubkin Mar 18 '14 at 22:45
• I agree with this answer, though I thought I would point to the one case I know where the "$gH = Hg$" definition makes things easier: When proving that a subgroup of index $2$ is normal. – Tobias Kildetoft Mar 26 '14 at 7:55
• I rather like the $gN = Ng$ characterisation; to me it reminds me of and is inspired by $g1 = 1g$, and drives the point home how N is identified with the identity in the quotient group. The same thing motivates ideals in rings. – Vandermonde Nov 4 '15 at 9:49

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.).

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

• Indeed, otherwise why do we care about normal subgroups at all, except that they are exactly kernels of homomorphisms? What would the attraction of the conditi on $gN=Ng$ be? :) – paul garrett Mar 18 '14 at 19:20
• @paulgarrett, I think that Ittay Weiss says it very well above (matheducators.stackexchange.com/a/249/2070); thinking in terms of equal right and left cosets makes it natural to multiply cosets, thus allowing us to manufacture new groups rather than requiring them already to exist as targets for our homomorphisms. – LSpice Sep 6 '14 at 7:07
• @LSpice, sure, yes, we can "manufacture" such groups, but this is too artifactual for my taste. To my taste, it would be more wholesome to characterize quotients "categorically", and then see how to prove existence by construction, rather than just do the construction. Set-theoretic constructions have lost some of their charm for me. – paul garrett Sep 6 '14 at 15:10

I like Gowers' fake history of normal subgroups.

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:

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.

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.

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.

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.