Previous: [[November Blog writing challenge]] Free groups are a construction that is actually quite natural, but don't often show up in a first course in abstract algebra. This isn't too surprising; some of the things you can do with them are quite delicate. Still, they offer a cool perspective on things. In group theory we often talk about a group using **generators** and **relations**. The generators of a group $G$ are a subset $S$ of elements such that any element in $G$ can be expressed as a finite combination of them. Conversely, the group generated by a finite combinations of elements $S$ in a group $G$ is called the **subgroup generated by $S$** and denoted $\langle S \rangle$. Relations, on the other hand, are equations in the generators that allows us to convert between equivalent finite combinations. This definition sometimes seems a little far-fetched, but really this is a very natural idea that comes up when you talk about groups. The classical example of talking about a group using generators and relations is the Dihedral group $D_{2n}$ which has the **presentation** $\langle r, s|r^n = s^2 = 1, rs = sr^{-1}\rangle$ Thinking about the group as a symmetry group of an $n$-gon, this is saying that the group is generated by the rotation $r$ that maps a vertex to the next one (i.e. a rotation by $\frac{2\pi}n$) and by reflecting the polygon across one of the lines of symmetry. The equations tell us that a full rotation or two flips get us back the original polygon, and the last equation says that flipping and rotating is the same as doing the opposite rotation and then flipping. If you've taken abstract algebra, none of this is new. Free groups arise when you start asking questions about this form of writing down groups. First, *Can we write every group in this form*? Thinking about it a little, the answer is yes: For any group $G$, take *every element in the group* as a generator and let our relations be the Cayley table of the group i.e. append all relations of the form $g \cdot h = gh$. This doesn't tell us very much about anything, really, except that a presentation for a group always exists. To make our lives easier, we'll now talk about finitely generated groups, which are groups which have a finite set of generators. Now that we know that presentations always exist, we can ask questions about them. For a given group $G$, is there a *smallest* presentation that gives it^[see [[Minimal presentations of groups and the Frattini subgroup|this sidenote though]]]? Can we always figure out a group from its presentation? Tell when two presentations are the same? These questions turn out to be surprisingly hard! In fact, the [word problem for groups](https://en.wikipedia.org/wiki/Word_problem_for_groups) which is the problem of determining if two *strings* (in the *same* presentation of a *single* group) represent the same element, is in general undecidable! Not conceding defeat, let's try to make our lives as simple as possible. What happens if we have *no* relations? If we have just one generator, and no relations, then the group $G = \langle x\rangle$ is just $\mathbb Z$. This is because there is no equation to ever kill off the $x$, so we get all its powers $x^n$ and an isomorphism $x \mapsto 1$ from $G$ to $\mathbb Z$. The next step up in complexity is having two generators and no relations. This group is called the free group on two generators: $F_2 = \langle a, b \rangle $ It has a pretty simple intuitive description: It is all strings from the alphabet $\{a,b,a^{-1}, b^{-1}\}$ in which whenever something and its inverse are next to each other, they cancel out: $aa^{-1}b = b$. But if there is nothing cancelable, something like $ab$ cannot be reduced to anything else because we have no other relations. Note that this is different from the free *abelian* group on two generators, $\mathbb Z^2 = \langle a, b|ab = ba\rangle$ which is just $\mathbb Z^2$ because we can rearrange any term into $a^n b^m$ for integers $n$ and $m$. We can't do that in $F_2$; $abab$ is not the same as $baba$. In a similar vein we can describe $F_2, F_3,\ldots$ and even $F_\infty$ on countably-infinitely many generators. But it is really $F_2$ that is the most interesting, and we will see more about that later. One very interesting thing about free groups, which we'll talk about more later, comes up when you think about maps out of free groups. If $\varphi: F_2 \to G$ is any group homomorphism, then it is described totally by the image $\varphi(a)$ and $\varphi(b)$, because those two generate the whole group. This is true for any set of generators for any groups. What's more unique about free groups is that we have total choice in picking $\varphi(a)$ and $\varphi(b$); in a group where there are relations the images will have to satisfy some equation, but we are *free* of any such constraints here! Next: [[Quotients of free groups]]