>"As an example we shall show that the dihedral group $D_n$ (...) is defined by the relations $$\tag {33} x^n,y^2,xyxy$$
>in the free group generated by $x,y$. It is clear that $D_n$ is generated by the rotation $R$ through an angle of $2\pi/n$ and the reflection $S$ in the $x$-axis. We have the relations $$\tag{34} R^n=1,S^2=1,SRS=R^{-1}$$

>Hence $D_n$ is a homomorphic image of $FG^{(2)}/K$ where $K$ is the normal subgroup generated by the elements $(33)$. We shall now show that $|FG^{(2)}/K| \leq 2n$ which will imply that $D_n\simeq FG^{(2)}/K$. Let $\bar x=xK$,$\bar y=yK$ (...). Then, s…

>"A group $G$ is said to be finitely generated if it contains a finite group of generators $\{a_1;1\leq a_i\leq r\}$. Then we have the homomorphism $\eta$ of $FG^{(r)}$ sending $x_i\to a_i$. Since the $a_i$ generate $G$, this is an epimorphism and $G\simeq FG^{(r)}/K$ where $K$ is the kernel of $\eta$. The normal subgroup $K$ is called the set of relations connecting the generators $a_i$. If $S$ is a subset of a group, we can define the normal subgroup generated by $S$ to be the intersection of all normal subgroups of the group containing $S$. If $S$ is a subset of $FG^{(r)}$ we say that $G…