Previous: [[Free groups]] The free group is very "large" in a sense, because the lack of relations disallows us from cutting its size. Turns out that there are a lot of interesting smaller things hidden inside it. The standard ways of getting a smaller group from a group are *subgroups* and *quotients*. Normally, subgroups are much easier to talk about, but in the case of free, we'll see that it's quotients that are more intuitive. Better still, this will let us talk about *relations* in a more formal manner. To recall, if we have a group $G$ and a *normal* subgroup $N$ of $G$, then the set of cosets $G/N$ has the structure of a group via $(gN)(hN) = (gh)N$ and is called the quotient group. In an intro group theory course one learns that the normality condition $gng^{-1} \in N \;\;\forall n \in N$ or $gNg^{-1} = N$ or $gN = Ng$ is precisely the condition that lets you turn the set of cosets into a group, but there is another way of looking at this. This comes from the fact that normal subgroups are precisely the kernels of homomorphisms. For this viewpoint, suppose we wanted to "forget" about the information of the normal subgroups, or "forcibly" set all elements in the normal subgroup to the identity. Then the new group constructed out of this procedure is the quotient group. To appreciate this viewpoint more, let's return to generators and relations. We saw in the last post that any group can be written using generators and relations. Suppose we have exactly that: $G = \langle S|R\rangle$ where $S$ is the set of generators and $R$ is the set of relations i.e. equations in $G$ that hold. We can in fact think of $R$ as a set of words in the group via the following observation. If we have any equation, for instance $ab = cb^3a,$ then we can move every element on the right to the left by multiplying by the inverse. In this case it will get us $aba^{-1}b^{-3}c^{-1} = e$ where $e$ is the identity element. So *every* relation is actually $\text{word} = e$ for some word in the group, and $R$ is just the set of words we have "forcibly" set to the identity. Now let $N(R)$ be the "smallest normal subgroup" generated by the relations $R$. This can be thought of as the subgroup of all elements that get sent to zero whenever $R$ gets sent to zero by any homomorphism $\phi:G \to H$ for any other group $H$. Or, we can construct this explicitly by constructing the general form of something in $R$ conjugated by a group element $G$, much like how we construct the subgroup generated by a set (exercise: write this down and prove it is actually a subgroup and that it actually normal; see a solution [here](https://www.physicsforums.com/threads/normal-subgroup-generated-by-a-subset-a.955664/)). Or we can take the set of all normal subgroups in $G$ and consider the intersection of all that contain $R$ as a subset. All of these are equivalent and give us the correct idea of "things that get zeroed out when $R$ gets zeroed out." The upshot of all this is this: For any group $G = \langle S|R\rangle$, we can express it as the equivalence class of words formed from symbols in $S$ (and the inverse symbols), while zeroing out all the words in $R$. We saw last time that the correct concept for "words formed from an alphabet" was the free group^[again, this is when we have inverses. If we don't have inverses it's actually the free *moinoid*] and the correct concept for "zeroing out elements" is quotienting by the normal subgroup generated by an element. Thus, $G = F(S)/N(R)$ i.e. a group presentation lets us construct $G$ as the quotient of a free group. Since we saw last time that every group has a group presentation, *every* group can be represented as the quotient of a free group! Next: [[The universal property of the free group]]