The symmetric group, 4

One of my views on teaching abstract algebra, with which my colleagues don’t all agree by any means, is that it is better to do rings before groups. The view for putting groups first is that they have just a single operation and four axioms, whereas rings are twice as complicated; so students can play with small examples of groups with more confidence that they really are groups.

I don’t think mathematics works that way, for two reasons.

Firstly, an important branch of mathematics doesn’t begin when some genius writes down some axioms plucked from her/his fertile brain, and begins to examine their consequences. In all significant cases, a subject was axiomatised when it already existed as a lively and important topic of study. Lagrange, Abel, and (especially) Galois knew very well what a group is, and proved important theorems about groups, long before von Dyck wrote down the axioms. (This example shows something else too. If you axiomatise an existing subject, you are obliged to do two things: firstly, to show that the objects of study in that field really do satisfy the axioms; and second, to indicate how much (if any) more general the axiomatic class is. In the case of groups, the second task is performed by Cayley’s Theorem: every axiomatic group is isomorphic to a group in the sense of Lagrange or Galois, that is, a transformation group or permutation group.)

Secondly, that is not how we learn mathematics either. Everything we learn has to be fitted in with, or grafted on to, what we already know. Now every mathematician certainly knows from earliest childhood about the natural numbers. Let us be charitable and suppose that we have all understood negative numbers from an early age as well, so that we know about the integers. Now my thesis is that there is one “canonical” example of a ring: the ring of integers.

The ring of integers is of course rather special: it is commutative, has an identity, has no zero-divisors, etc. Nevertheless, almost every important notion in elementary ring theory can be illustrated using the integers. (The ring of even integers has no identity; the ring of integers modulo n, for composite n, has zero-divisors.) More important still, these two notions illustrate the crucial concepts of subring and quotient ring.

So, if we start with ring theory, the students have a ready-made example of a ring, against which they can test their understanding. My opinion is that, with just a bit of care, it is a very good example indeed. A ring is “something like the integers, but possibly not commutative and possibly having zero divisors …” Of course, the idea of treating the ring of integers as a single object will be new to the students; but that is what algebra is about, so why compound the difficulty?

Is there a group that plays a similar role for group theory?

The fact that the integers form a commutative ring is not such a big handicap to understanding; the important basic ideas don’t depend on this so much. But, in group theory, they do; in abelian groups, why would one bother define conjugacy, distinguish between subgroups and normal subgroups, define the centre, commutators, etc.? So our prototypical group had better be non-abelian.

The smallest non-abelian group is the symmetric group S3, and that might be the best choice. But students will not be familiar with that group as such before taking the course. You have to say “consider the symmetries of an equilateral triangle” or something similar before starting, so that they have to learn about the typical group at the same time as they learn about groups in general. Or you could simply write down the 6×6 Cayley table and ask the students to take the associative law on trust …

Perhaps the ideal answer would be something like this. Students could be introduced to permutations in a basic discrete mathematics course, before they get to abstract algebra. They would learn that permutations don’t necessarily commute, that a permutation has an order (and how to calculate this), that they can be composed (this always makes a good hands-on exam question). Then they will be ready to meet the symmetric group, as a group, when they start learning algebra. Again, the students won’t have thought of the set of all permutations as a single object before; but again, this is what algebra is about.

By this stage, students probably also know that matrices don’t commute. But matrices can be added as well as multiplied; it is a little unnatural to learn about the general linear group as a group before you learn the matrix ring as a ring. (It is, simply, the group of units of the matrix ring.)

Also, as I am trying to show in this series, the symmetric group is an inexhaustible source of mathematical delight. Even in algebra, the subgroups and quotient groups of S4 give very good illustrations of these basic concepts. (If you want a slightly harder example, there is a homomorphism from GL(2,3) to S4; the inverse images of subgroups of S4 include the quaternion group in one of its simplest manifestations.)

So the symmetric group is my candidate for a prototypical group; but I still prefer to teach rings before groups.

As a footnote, what we refer to as “Lagrange’s Theorem” was proved by Lagrange just for the case of symmetric groups. Perhaps the mathematical community regard these groups as sufficiently generic that he deserves to have his name attached to the general theorem? (The arbitrariness of mathematical nomenclature makes me think that this is not actually the case!)


Footnote: Here is a problem in ring theory, from Ulam’s Problem Book.

Let R be a ring whose underlying set is the integers, and whose multiplication is the same as that in the integers (but the addition is unspecified). What can be said about R? In particular, show that R has characteristic 0 or 3, and give examples to show that both cases occur.

Contrary to appearances, the way to tackle this question is not with your bare hands, but using the concepts in chapter 1 of your favourite ring theory book. This is an unusual example of a situation where the abstract theory developed by mathematicians is exactly what is needed to solve an unlikely-looking problem!

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I count all the things that need to be counted.
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7 Responses to The symmetric group, 4

  1. Sam Tarzi says:

    Perhaps another reason as to why ring theory should be taught before group theory is its closer similarity to vector space theory, which often crops up in school mathematics curricula. As an extension of this, ring theory appears to have less rigid composition rules than group theory, so perhaps is an easier topic to grasp on a first encounter. I wonder if in the distant future, algebra could be introduced as universal algebra, with groups, rings, lattices and modules as examples, and where for instance the isomorphism theorems in universal algebra reduce to their equivalents for these structures?

    • When I arrived in Oxford, the first abstract algebra taught to the second-year students was category theory and groups acting on sets. I can see the reasons why this was done (indeed, for a given group G, the G-sets form a nice example of a category); but as you can imagine, it was not very well-received by the students!

  2. great post,but i experience some trouble in understanding the final paragraph, can you please explain a little bit indepth?

  3. A couple of people have asked, so here is the solution to the puzzle. We have a ring with the same multiplication as the integers, but we know nothing about the addition. Any property which depends only on multiplication and holds in the integers must also hold in the unknown ring. In particular, it is commutative; has an identity; is an integral domain (no zero-divisors); has only two units (the integers 1 and -1, though we don’t know yet that -1 is the negative of 1 in our unknown ring!); has a countably infinite set of irreducibles; and has unique factorisation.

    Conversely, a ring with all these properties is multiplicatively isomorphic to the ring of integers. (Enumerate the associate pairs of irreducibles, match them up arbitrarily with the integer primes; then use unique factorisation to match every element of the ring with an integer. It is almost obvious that multiplication is preserved.)

    Now our unknown ring, being an integral domain, has characteristic zero or prime. Suppose that it is a prime p. Then there are p–1 units. So p≤3. Suppose that p=2; temporarily use a for the non-identity unit. Then a^2=1, so (a–1)^2=0, contradicting the fact that the ring is an integral domain and a is not 1. So p=3.

    As an example, take the ring of polynomials over the field of three elements. By well-known theory, it is a unique factorisation domain. Its only units are 1 and -1, and Euclid’s argument shows that it has infinitely many primes.

  4. Pingback: The symmetric group, 3 « Peter Cameron's Blog

  5. Pingback: The symmetric group, 5 « Peter Cameron's Blog

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