Representative Pedagogical Examples of Groups, Real Functions, Modules, etc.

Question

In the preface of Munkres's Topology, he writes,

Fortunately, one does not need too many counterexamples for a first course; there is a fairly short list that will suffice for most purposes. Let me give it here:

$\mathbb{R}^J$ the product of the real line with itself, in the product, uniform, and box topologies.

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[3 more examples]

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These are the examples you should master and remember; they will be exploited again and again.

I found that having a short, fixed list of topological spaces that I could refer to and investigate whenever I learned a new idea was very helpful for learning point set topology.

Does anyone have similar pairings of a subject and a small list of representative examples to keep in mind while learning that subject? I enjoy returning to the same few examples every time because I feel that I get to know them very well as objects.

An example of the format I am looking for would be a list like the one below.

Basic Ring Theory

• $\mathbb{Z}, \mathbb{Q}$.
• $M_n(\mathbb{Z}), M_n(\mathbb{Q})$.
• etc.

Answer 1

Finite groups: The cyclic groups; the dihedral groups; the symmetric groups; the alternating groups; the quaternion group.

Answer 2

Graph Theory: the complete graphs; the complete bipartite graphs; trees; the Petersen graph.