----- > [!proposition] Proposition. ([[conjugacy classes of the dihedral group]]) > Denote by $D_{n}$ the [[dihedral group]]. Its [[conjugate|conjugacy classes]] are... > [!proof]- Proof. ([[conjugacy classes of the dihedral group]]) > # Case when $n$ is *even* > Clearly $[e]=\{ e \}$, and from [[conjugate#^e21140|conjugacy class is singleton iff element belongs to center]] and [[center of a group#^accce0|center of dihedral group]], $[x^{n/2}]=\{ x^{n/2} \}$. Next we move to classify the reflections (elements of the form $yx^{k}$). Observe that the [[conjugate|conjugation]] of $yx^{k}$ by an arbitrary element in $D_{n}$ takes the form $1) \ \ \ (x ^{\ell}) yx^{k} (x^{- \ell}) \text{ (conj. by a rotation)}$ > or $2) \ \ \ (yx^{\ell})yx^{k}(x^{- \ell }y) \ \text{(conj. by a reflection)}.$ > Also recall that in general, $yx^{\ell}=x^{-\ell}y$ for this will be employed in the computations to follow. > > Analyzing $(1)$ we find $\begin{align} > x^{\ell} yx^{k} x^{- \ell}= & x^{\ell }yx^{k-\ell} \\ > = & x^{2 \ell - k}y \\ > = & yx^{k-2 \ell}, > \end{align}$ > which reading from bottom-to-top tells us that an arbitrary $yx^{k}$ is [[conjugate]] to $yx^{k-2 \ell}$ (confirming intuition that there are two 'types' of reflection— over sides and over corners). Analyzing $(2)$ we find $\begin{align} > (yx^{\ell})yx^{k} (x^{-\ell}y)= & yx^{\ell}yx^{k}yx^{\ell} \\ > = & y^{2} x^{\ell-k}yx^{\ell} \\ > = & y^{3} x^{k-\ell}x ^{\ell} \\ > = & yx^{k}, > \end{align}$ > implying that we cannot reach any element in $D_{n}$ other than $yx^{k}$ itself when we [[conjugate]] $yx^{k}$ by a reflection. Therefore, for any $d\in D_{n}$$yx^{k} \sim d \iff d \in \{ yx^{k-2 \ell \text{ mod }n} : \ell \in \mathbb{Z}_{}\}.$ > > Finally, we classify the rotations other than $x^{n/2}$. Let $x^{k}$ be an arbitrary rotation. The [[conjugate|conjugation]] of $x^{k}$ by an arbitrary element in $D_{n}$ takes the form $1) \ \ \ (x ^{\ell}) x^{k} (x^{- \ell}) \text{ (conj. by a rotation)}$ > or $2) \ \ \ (yx^{\ell})x^{k}(x^{- \ell }y) \ \text{(conj. by a reflection)}.$ > It is clear that $(1)=x^{k}$. So, we cannot reach any element in $D_{n}$ other than $x^{k}$ itself when we [[conjugate]] $x^{k}$ by a reflection. Analyzing case $(2)$ we obtain $\begin{align} > yx^{\ell}x^{k}x^{-\ell}y = yx^{k}y=x^{-k} > \end{align},$ > telling us that for any $d \in D_{n}$ $x^{k} \sim d \iff d=x^{n-k}.$ > From this we conclude that when $n$ *is even*, the [[conjugate|conjugacy classes]] of $D_{n}$ are $\{ e \}, \{ x^{n/2} \}, \{ yx ^{k} : k \text{ is odd} \}, \{ yx^{k}: k \text{ is even} \}, \{ x, x^{n-1}\}, \{ x^{2}, x^{n-2} \},\dots, \{ x^{n/2-1}, x^{n/2+1} \}.$ > In this case the [[class equation]] is $|D_{n}|=|Z(G)|+\sum_{[x]: x \notin Z(G)}^{} |[x]|=2+ n + \overbrace{2 + 2 + \dots +2}^{ (n-2)/2\text{ times}}.$ > > # Case when $n$ *is odd* > Clearly $[e]=\{ e \}$. Next we move to classify the reflections — i.e., elements in $D_{n}$ of the form $yx^{k}$. By considering the [[conjugate|conjugation]] of $yx^{k}$ by an arbitrary element of $D_{n}$ we will be able to see when conditions are required by $d \in D_{n}$ to have $d \sim D_{n}$. There are two cases: conjugation by reflection and by rotation. The latter case proceeds identically to as it did when $n$ was even: we cannot reach any element in $D_{n}$ other than $yx^{k}$ itself when we [[conjugate]] $yx^{k}$ by a reflection. For the rotation by $x^{\ell}$ case, we may again repeat the work before to obtain $yx^{k} \sim d \iff d \in \{ yx^{k-2 \ell \text{ mod }n} : \ell \in \mathbb{Z}_{}\},$ > but in this case there is no splitting based in the parity of $k$ because $n$ is odd. From this we conclude that when $n$ *is odd*, the [[conjugate|conjugacy classes]] of $D_{n}$ are $\{ e \}, \{ yx ^{k} : k \in \mathbb{Z} \}, \{ x, x^{n-1}\},\{ x^{2}, x^{n-2} \}, \dots, \{ x^{\text{floor } n/2 }, x ^{\text{ceil }n/2 } \}$. > > > The [[class equation]] is hence $|D_{n}|=|Z(G)|+\sum_{[x]: x \notin Z(G)}^{} |[x]|=1+ n + \overbrace{2 + 2 + \dots +2}^{\text{floor }n/2 = (n-1)/2\text{ times}}.$ > > > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```