Presentations of the dihedral and quaternion groups I am reading Lang’s algebra text and he is currently discussion generators of groups and gives the following examples: 
Is Lang suggesting here that any two elements of either group which satisfy those relations will generate the group? Or is he instead simply saying that(referring to the dihedral group) the clockwise rotation and reflection about two vertices are generators for the group and that they satisfy those relations(that is, he is alluding to specific generators and stating relations they satisfy)?
Note: I understand that he is using typical notation for the quaternion group but it is easy to see that any pair of quaternions which satisfy those relations will generate the group as well.
 A: In addition to Gerry Myerson's comments, I would like to offer another perspective: what I personally would understand by reading this is that Lang is defining a presentation of a group and defining it to be $D_8$ or $Q_8$.
What I mean by this that, in the case of the dihedral group, we ignore any visualizations with the square - just take a generating set $\{\sigma, \tau\}$ and form the group $\langle \sigma,\tau\mid \sigma^4=e,\tau^2=e,\tau\sigma\tau^{-1}=\sigma^3\rangle$, which is a definition of some group, which we call "the group of symmetries of the square", or $D_8$. The same applies for the quaternions.
Conversely, once we define $D_8,Q_8$ like this, we cannot take any old elements $\sigma',\tau'\in D_8$ that satisfy the relations and claim that they generate the group - what if $\sigma'=\tau'=e$? We would need to additionally require that $\sigma',\tau'$ generate a group of order 8.
A: It might be well to recall the way that presentations of groups work.  The presentation  $\langle S|R\rangle $ refers to the group which is the quotient of the free group on the generators, the elements of  $S $, by the normal subgroup,  or the so-called normal closure, of the subgroup generated by the relations $R $.
