Group Theory: A Powerful Tool for Real World Solutions

[matqkks]

What is the most motivating way to introduce group theory to first year undergraduate students? I am looking for some real life motivation or something which has a real impact.

 

[Deveno]

Well, there's a number of possible approaches, depending on the things you want to emphasize.

One method is to emphasize the combinatorial aspects, using permutations, and theorems like Burnside's Lemma to examine arrangements and colorings. One can broaden this approach to examining other examples of symmetry such as rotational (cyclic groups), planar (wallpaper and frieze groups), and braid groups. The nice thing about these aspects of group theory is you can find plentiful "real-life" examples.

One can alternatively take the "solving equations" path, by showing groups are the logical "distillation" of the methods used in high-school algebra, without the somewhat artificial constraint of commutativity (sometimes, maybe even most of the time, the order in which we do things matters). The simple equation $ax = b$ is at the heart of a great deal of mathematics ($a$ and $b$ might be matrices, for example). Seeing group elements as *operators* which act on other group elements by the group operation is something that pays off in differential geometry, and the more advanced study of Lie groups and Lie algebras.

Groups are a good introduction to the study of "structure" itself-the constructions investigated in groups, such as subgroups, kernels, quotients, and the direct product will appear again in different guises in linear algebra, ring theory, functions spaces, and other branches of mathematics. It's good practice for people to see what they can prove-*just from the rules given*. The notion of "play" should not be under-estimated, here-with some groups, the objects are simple enough people can *explore* what happens, much like exploring the rooms of a new house they've never been in.

Most of your students have *already been using groups*, they just don't know it, because the names haven't been named. Exponential maps and logarithms are homomorphisms. The circle group (complex numbers of magnitude 1) is what trigonometry is all about. Modular arithmetic is an example of quotient groups in action. Rolling a six-sided die is the rotation group of the cube in action. A change of basis for a square matrix is an example of conjugation. Complex conjugation and the identity map are an automorphism group of order 2.

The "simplest" approach may well be the "symmetry" angle I outlined above. For example, you can ask your students the following questions:

Suppose I have a rectangular bed, with a fitted sheet that has an "outside" (the side that goes up-maybe it's smooth, and comfy), and an "inside" (it's rough, and not so comfy). The sheet has a tag on the inside in one corner. How many corners can the tag go on?

How many corners can it go on, if the sheet is reversible?

These are questions of great importance to anyone who has to make a bed-and also an illustration of the Klein viergruppe.

The idea is, one should not be scared of these "group things", they're actually very straight-forward. Much easier than say, Lebesgue integrals.