for all \(n\)-tuples \(\boldx=(x_1,x_2,\dots, x_n)\) and \(\boldy=(y_1,y_2,\dots, y_n)\text{.}\) An isometry (or rigid motion) of \(\R^n\) with respect to \(d\) is a function \(f\colon \R^n\rightarrow \R^n\) satisfying
Rotations about a point, reflections through a line, and translation by a fixed vector \(\boldv_0\) are easily seen to be examples of rigid motions of the plane. Furthermore, it can be shown that a rigid motion \(f\colon \R^2\rightarrow \R^2\) that maps the origin \(\boldzero=(0,0)\) to itself is a linear transformation, that itself is either rotation about \(\boldzero\) or reflection through a line passing through \(\boldzero\text{.}\) It follows from this that any rigid motion \(f\colon \R^2\rightarrow \R^2\) has a formula of the form
Fix an integer \(n\geq 3\text{.}\) Let \(P_n\subseteq \R^2\) be the regular \(n\)-gon centered at the origin with one vertex at \((1,0)\text{.}\) We define \(D_n\) to be the set of rigid motions of \(\R^2\) that map \(P_n\) onto itself: i.e.,
The group identity of \(D_n\) is the identity function \(\id\colon \R^2\rightarrow \R^2\text{.}\) Given an element \(f\in D_n\text{,}\) its group inverse is its function inverse \(f^{-1}\text{.}\)
What we call \(D_{n}\) the book calls \(D_{2n}\text{!}\) This is somewhat unfortunate, but it turns out there is no real consensus in the mathematical community as to which is the preferred notation. As we see below, the \(2n\) subscript has the advantage of indicating the cardinality of the group in question. On the other hand, the \(n\) subscript reminds us that the group is defined in terms of isometries that fix an \(n\)-gon. The latter is consistent with our notation for permutation groups \(S_n\) (to be introduced soon), which are defined as the group of permutations acting on a set of \(n\) elements. That fact, along with the fact that I was simply raised on \(D_n\) and not \(D_{2n}\) has persuaded me to part ways with the text in this case.
Fix an integer \(n\geq 3\text{,}\) let \(P_n\) be the regular \(n\)-gon centered at the origin with one vertex at \((1,0)\text{,}\) and let \(D_n\) be the corresponding dihedral group.
Furthermore, define \(r\colon \R^2\rightarrow \R^2\) to be rotation by \(2\pi/n\) about the origin, and define \(s\colon \R^2\rightarrow \R^2\) to be reflection across the \(x\)-axis.
For each element \(r^is^j\in D_n\text{,}\) where \(r\) and \(s\) are defined as in Theorem 1.4.5, give an explicit geometric description of this rigid motion.
In other words, in the description of \(D_n\) given in Theorem 1.4.5, we can replace the reflection \(s\) with any of the reflections \(s'=r^is\text{.}\)
together with the fact that \(D_n=\{r^is^j\mid 0\leq i\leq n-1, 0\leq j\leq 1\}\) are essentially all we need to know to be able to do computations in the dihedral group \(D_n\text{.}\)Group presentation notation gives us a nice way of summarizing this defining properties. A group presentation of \(D_n\) is
Although we introduce the notation here as a convenient way to summarize how an established group, namely \(D_n\text{,}\) is “generated” by the elements \(r\) and \(s\text{,}\) the notation can also be used to construct groups from scratch. Put another way, any presentation like the one above can be shown to be the presentation of an actual existing group. For example, the presentation
can be shown to be the define a certain group \(G\) of cardinality \(4\text{.}\) As it turns out, to make good sense of this construction, a significant amount of theory must be mobilized. As such, we will be careful to confine ourselves to using the presentation notation only to as a summary description of a group we know already to exist. The quaternion group below is a good example of this.
