What are the "best" groups to use as examples while learning new concepts in algebra?

Question

This question was asked to Math SE at first but it seems like it is more appropriate to ask it here.

While learning new concepts in algebra it is quite helpful to check some examples which includes the groups that we are familiar with, like $\mathbb{Z}/6\mathbb{Z}$, $S_n$, $GL(2,\mathbb{R}), \dots$

For example, while learning quotient groups, we can choose a normal subgroup $H=\{\bar{0},\bar{2},\bar{4}\}$ of the group $\mathbb{Z}/6\mathbb{Z}$ to understand what is going on.

I know my question seems subjective, but, I think that some of you have the groups that you know well and use them often. So, what groups -or rings, fields- can you suggest? Thanks!

Answer 1

Symmetry groups of basic regular polygons and polyhedra. For one thing, they are noncommutative; for another, sometimes they coincide with other "known" groups; for a third, there can be physical meaning to some concepts in group theory.

And they are very "hands-on" - it can be fun to make sure students understand how to generate the symmetry group for the cube with as few elements as possible by building a toothpick model. (This kind of thing can even work with non-major classes, if you wanted to introduce them to group theory - presumably somewhat more slowly than for an upper-level US group theory course.)

Answer 2

You mentioned the linear groups over the real numbers, which I think is a useful example, with the potential drawback that it is infinite.

To "remedy" this one can consider the linear groups over finite fields and even rings, yet for concreteness just over $\mathbb{Z}/n\mathbb{Z}$ for some particular small $n$ should suffice for a start.

Computing with low-dimensional matrices is relatively efficient, and the fact that one is dealing with matrices gives some obvious structure to the group.

There are plenty of subgroups (diagonal matrices, upper triangular matrices, triangular with all ones on the diagonal, the special linear group, etc). Some of them normal, some not.

To determine the center of these groups is manageable but still interesting.

The isomorphism of $S_3$ and $GL(2, \mathbb{Z}/2\mathbb{Z})$ is curious.

Answer 3

In addition to the dihedral groups, the quaternion group $Q_8$, the symmetric and alternating groups, and small matrix groups, I have always found the following examples useful:

Automorphism groups of graphs: I usually introduce these at around the same time I introduce permutations. This gives a wealth of interesting examples of groups, and also gives a simple example of how groups can be used to understand other mathematical structures. There are lots of small graphs (e.g. trees) that provide simple examples of quotients and normal subgroups, and you can also use graphs to give nontrivial applications of various group action concepts. For example, I have a homework problem where students compute the order of the automorphism group of the Petersen graph using the orbit-stabilizer theorem.

Unit groups: Gallian's book places an emphasis on the unit groups $U(n)$, i.e. the group of units in the integers modulo $n$. I really like the emphasis on these groups, because their structure is not obvious, so students really have to dig into them to figure them out. For example, listing the subgroups of $U(15)$ or $U(24)$ is an interesting exercise, as is computing the automorphisms of either of these.

Symmetry groups of polyhedra: Others have already mentioned these so I won't go into too much detail here, but I should point out that lots of polyhedra beyond the platonic solids and prisms have interesting symmetry groups. For example, one of my homework problems asks the students to compute the symmetry group of the tetrahedron with vertices at $(8,8,1)$, $(8,-8,-1)$, $(-8,8,-1)$, and $(-8,-8,1)$. I'm also fond of antiprisms.