§7 Elliptic Curves

Definition 7.1 Let $f(x)$ be a well-behaved cubic polynomial. The elliptic curve defined by $f$ is the set

§ 6 Free Groups

One way to define a group is to specify a collection of generators together with a collection of relations satisfied by those generators.

Question: What does a group with a set of generators but no relations look like? If the set of generators is $S$, such a group is called the free group on $S$.

§ 5 Cycle Notation

Definition 5.1 Suppose we have a group action

on a set $X$. Fix $g \in G$. We call the action

the action of $g$ on $X$.

Here $\langle g \rangle \to G$ is a group homomorphism because the inclusion map of a subgroup is a group homomorphism. The map $\langle g \rangle \to \mathrm{Aut}_{\text{set}}$ is a group homomorphism because the composition of two group homomorphisms is again a group homomorphism.

Essentially, the action of $g$ on $X$ can be expressed using cycle notation by decomposing $g$ into cycles, where each cycle corresponds to an orbit of the action of $g$ on $X$.

§4 Group Actions

§4.1 Motivation for Group Actions

As one of the greatest mathematicians once said, let mathematics speak for itself—you should not feel that everything needs to be motivated. If something is beautiful, it motivates itself. That being said, I do not really want to motivate group actions for you, but their history is actually quite interesting, so it is worth discussing.

Suppose you were a French or German mathematician in the middle of the nineteenth century. Your definition of a group would not have been the one we gave above. In fact, a group was essentially understood simply as a matrix group $GL_n(\mathbb{R})$, or perhaps over $\mathbb{C}$ if you wanted to work in a world where everything was especially beautiful. Naturally, we have a bijection

where $V$ is a real vector space of dimension $n$.

Thus, rather than blindly performing row reduction, it is natural to study how elements of $GL_n(\mathbb{R})$ behave through the bijection above. The "group action" appearing in the automorphism description (4.1) actually gives rise to the more abstract theory of linear algebra that you have already studied! This is what is called a group representation: whenever you have a group homomorphism from $G$ to the group of linear automorphisms of some vector space, you have a group representation, which we will study shortly.

The important point is that group actions arise naturally from studying the automorphisms in (4.1), so what we are studying is not an entirely artificial construction. To understand how we move from studying linear automorphisms of vector spaces to studying automorphisms of sets, note that the notion of a group action occurs universally outside linear algebra as well. So why not "generalize" the theory of group actions from vector spaces to sets?

This type of process turns out to be extremely important in algebra. You look at what structure you have, try to remove whatever structure is not actually necessary for studying the "abstract theory", and then see whether an interesting theory remains. It was essentially this process that led Emmy Nöther, in the early twentieth century, toward the modern abstract notion of a group.

§3 Maps of Groups

Whenever you define a new idea, it is useful to know what kinds of functions are naturally associated with it.

Question: What types of functions should we study?

Example 3.1