4.2: Dihedral Groups
- Page ID
- 97997
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We can think of finite cyclic groups as groups that describe rotational symmetry. In particular, \(R_n\) is the group of rotational symmetries of a regular \(n\)-gon. Dihedral groups are those groups that describe both rotational and reflectional symmetry of regular \(n\)-gons.
For \(n\geq 3\), the dihedral group \(D_n\) is defined to be the group consisting of the symmetry actions of a regular \(n\)-gon, where the operation is composition of actions.
For example, as we’ve seen, \(D_3\) and \(D_4\) are the symmetry groups of equilateral triangles and squares, respectively. The symmetry group of a regular pentagon is denoted by \(D_5\). It is a well-known fact from geometry that the composition of two reflections in the plane is a rotation by twice the angle between the reflecting lines.
The group \(D_n\) is a non-abelian group of order \(2n\).
Fix \(n\geq 3\) and consider \(D_n\). Let \(r\) be rotation clockwise by \(360^{\circ}/n\) and let \(s\) and \(s'\) be any two adjacent reflections of a regular \(n\)-gon. Then
- \(D_n=\langle r,s\rangle =\{\underbrace{e,r,r^2,\ldots, r^{n-1}}_{\text{rotations}},\underbrace{s,sr,sr^2,\ldots,sr^{n-1}}_{\text{reflections}}\}\) and
- \(D_n=\langle s,s'\rangle = \text{all possible products of }s\text{ and }s'\).
The next result is an obvious corollary of Theorem \(\PageIndex{2}\).
For \(n\geq 3\), \(R_n\leq D_n\).
The following theorem generalizes many of the relations we have witnessed in the Cayley diagrams for the dihedral groups \(D_3\) and \(D_4\).
Fix \(n\geq 3\) and consider \(D_n\). Let \(r\) be rotation clockwise by \(360^{\circ}/n\) and let \(s\) and \(s'\) be any two adjacent reflections of a regular \(n\)-gon. Then the following relations hold.
- \(r^n = s^2 = (s')^2 =e\),
- \(r^{-k} = r^{n-k}\) (special case: \(r^{-1}=r^{n-1}\)),
- \(sr^k=r^{n-k}s\) (special case: \(sr=r^{n-1}s\)),
- \(\underbrace{ss's\cdots}_{n\text{ factors}}=\underbrace{s'ss'\cdots}_{n\text{ factors}}\).
From Theorem \(\PageIndex{2}\), we know \[D_n=\langle r,s\rangle =\{\underbrace{e,r,r^2,\ldots, r^{n-1}}_{\text{rotations}},\underbrace{s,sr,sr^2,\ldots,sr^{n-1}}_{\text{reflections}}\}.\] If you were to create the group table for \(D_n\) so that the rows and columns of the table were labeled by \(e,r,r^2,\ldots, r^{n-1},s,sr,sr^2,\ldots,sr^{n-1}\) (in exactly that order), do any patterns arise? Where are the rotations? Where are the reflections?
What does the Cayley diagram for \(D_n\) look like if we use \(\{r,s\}\) as the generating set? What if we use \(\{s,s'\}\) as the generating set?