TITLE: Understanding group presentation as a quotient
QUESTION [6 upvotes]: I'm just starting to learn a little group theory, so please forgive any ignorance I demonstrate in the following.
I'm trying to understand the concept of a group being defined based on its presentation $G = \langle S \mid R \rangle$ as the quotient group of the free group $F_S$ generated by $S$, and the normal subgroup $R^{F_S}$ generated by $R$ in $F_S$.
I understand the basic idea that the normal subgroup $R^{F_S}$, is gathering up all the elements of $F_S$ that are to be defined as equivalent to $1_G$ (the identity in group $G$). I also understand that dividing $F_S$ by $R^{F_S}$ (is "dividing" the correct term?) partitions $F_S$ and that each of these partitions will essentially end up as a different element of $G$, and further, since $R^{F_S}$ is a normal subgroup, the partitions can be turned into a group by inheriting the group operation of $F_S$.
But I'm having trouble wrapping my head around exactly how the relators set $R$ partitions $F_S$. Not mechanically, that part I understand in terms of finding the cosets $gR^{F_S}$, but in a more intuitive sense, how do we tell how these relators define the structure of the group $G$?
Taking the dihedral group $D_4$, for instance, presenting it as $\langle r, v | r^4 = v^2 = rfrf = 1 \rangle$, how do we understand from this that $fR^{F_S}$, and $r^3R^{F_S}$, and $rfrR^{F_S}$, will all produce their "own" cosets, instead of being equal to $R^{F_S}$, or equal to one another?
My gut feel is that it has something to do with being "less" than the relators, or in some sense "comprime" to the relators, but I don't think I have the vocabulary to describe it correctly, and I may, for that matter, be completely off base.

REPLY [4 votes]: Jack Schmidt's answer is about how to compute effectively with group presentations, so let me try and give a more intuitive or conceptual answer.
The first point to note is that, although the definition is "correct", it is not particularly helpful - some definitions are like that. People who work a lot with group presentations do not typically think in terms of normal closures of sets of defining relators. When teaching introductory courses on group theory, I introduce presentations very early, because they are so useful for giving precise but compact descriptions of small examples, but I do this even before the students have encountered quotient groups (let alone free groups), so there is no question of giving the formal definition.
I define the group $\langle X \mid R \rangle$, where $X$ is a set of generators, and $R$ is usually given as a set of equations among words in the generators (like $\langle x,y \mid x^2=y^3=(xy)^2=1 \rangle$) as the "largest" group that is generated by $X$ in which the equations in $R$ hold. Of course I point out that I have not said what "largest" means for infinite groups, and I have not attempted to justify the implied claim that there is always a unique largest such group, but I give some examples (such as the one above, which defines the dihedral group of order 6), in which it can easily be proved that there is a unique largest such group.
Although the above definition is not formal, I think it expresses intuitively what we are trying to define. From the formal definition of $G = \langle X \mid R \rangle$, as a quotient $F(X)/\langle R \rangle^{F(X)}$ of a free group by the normal closure of the defining relators, you can go on to prove some basic properties such as
      
 The group $G$ is generated by the images in the quotient group of the generators $X$ of the free group. It is customary to abuse notation and denote these generators also by $X$, although you need to be aware that two elements of the original $X$ could have the same image in $G$.
 There is a fundamental result expressing the universal property of $G$: if $H$ is any group, and $\phi:X \to H$ is any map with the property that the images of the elements of $R$ under $\phi$ (you need to say exactly what that means) are all equal to the identity in $H$, then $\phi$ extends uniquely to a group homomorphism $G \to H$. This is not hard to prove from the definition of $G$.
 You should learn the basic Tietze transformations for manipulating group presentations. Again, the proof that these work can be proved easily from the definition.

I have found that once you are familiar with the above three properties, there is rarely any need to go back the definition of $G$ as a quotient of a free group.