---- > [!definition] Definition. ([[conjugate]]) > Let $G$ be a [[group]] and let $g,h \in G$. We call the element $ghg^{-1}$ the **conjugate of $h$ by $g$**. > \ > For $x,y \in G$, we say that **$x$ and $y$ are conjugate in $G$** if there exists $g \in G$ such that $y=gxg^{-1}$. We say this is **witnessed** by $g$. > \ > Conjugacy is an [[equivalence relation]]; the [[equivalence class]]es it creates are called **conjugacy classes** and denoted $[x]$ or $C_{x}$. > [!note] > Recall that, viewing $G$ and $H$ as [[group#^equivalence|one-object categories]], a [[group homomorphism]] $G \to H$ is just a [[covariant functor|functor]] $G \to \mathsf{Grp}$. > Given two [[group homomorphism|group homomorphisms]] $u,v: G \to H$, a [[natural transformation]] $u \to v$ is the data of some $h \in H$ satisfying $hu(g)=v(g)h$ for all $g \in G$, i.e., a witness that $u$ and $v$ are conjugate homomorphisms. ^note > [!basicproperties] > - [[conjugate#^bab0ac|conjugacy is an equivalence relation]] > - conjugacy classes of [[abelian group]]s are singletons > - $C_{x}=\{ x \}$ iff $x$ belongs to the [[center of a group|center]] of $G$ > - Order of elements in conjugacy class is constant. (how about the converse?) > - If $G$ is a finite [[group]], [[size of conjugacy class divides order of finite group|the size of any conjugacy class in]] $G$ divides $|G|$ ^a1150e > [!proof] Proof of Basic Properties. > Write $y \sim x$ if there exists $g \in G$ such that $y=gxg^{-1}$. $\sim$ is reflexive because $y = eye^{-1} \implies y \sim y$. $\sim$ is symmetric because $y \sim x \implies y=gxg^{-1} \implies x =g ^{-1} y g \implies x \sim y$. To see $\sim$ is transitive, suppose that $x \sim y$ and $y \sim z$. Write $x=gyg^{-1}=g(hzh^{-1})g^{-1}=(gh)z(gh)^{-1}.$ > \ > To see that conjugacy classes of [[abelian group]]s are singletons, let $G$ be an abelian group and let $x,y \in G$ with $x \sim y$. Then $\ex g \in G$ s.t. $y=gxg^{-1}$, but $gxg^{-1}=g(g^{-1} x)=e$. So $y=x$ and hence $[x]=\{ x \}$. > \ > To see that $C_{x}=\{ x \}$ iff $x \in Z(G)$, note that if $C_{x}=[x]$, then for a > \ > To see that conjugacy classes contain elements of identical [[order of a group|order]] in $G$, let $x,y$ be in the same conjugacy class and let $x$ have [[order of a group|order]] $n$. So, $y=gxg^{-1}$ for some $g \in G$. We have $y^{n}=(gxg^{-1})^{n}=gx^{n}g^{-1}=geg^{-1}=e,$ > thus $y$ also has order $n$. > \ > Also see [[size of conjugacy class divides order of finite group]] ^bab0ac > [!intuition] > **Perspective 1**. In light of the properties above, you can think about conjugacy classes as capturing how 'abelian' a [[group]] is: the fewer there are, the 'less abelian' the group is. > \ > **Perspective 2**. Recall the notion of [[similar|similar matrices]]: $A,B \in \mathbb{F}^{n \times n}$ are **similar** $A=SBS ^{-1}$ for some $S \in \mathbb{F}^{n \times n}$. That is to say, $A$ and $B$ are similar if they represent the same [[linear operator]]; if $A=S$ under a [[change of basis formula|change of basis formula]]. [[conjugate|Conjugation]] in [[group|group theory]] should be conceptualized similarly: two elements are [[conjugate]] in $G$ if they represent the same 'action' up to a 'change in perspective'. For example, in $D_{4}$ horizontal and vertical reflection is the same 'action', for this is witnessed by the change in perspective offered by $180^{\circ}$ rotation. In symbol, this allows us to immediately ascertain that in $D_{4}$ $\sigma \sim \sigma \tau^{2}$ as witnessed by $\tau^{2}$: $\sigma = (\tau^{2})^{-1} (\sigma \tau^{2}) (\tau^{2})$. > \ > **Perspectives 1 + 2**. [[TODO]] ^intuition > [!basicexample] Example. (Conjugacy classes in $S_{3}$) > Let $S_{3}$ denote the [[symmetric group]] on $3$ letters. What are its conjugacy ([[equivalence class|equivalence]]) classes? > \ > First consider $\{ e \}$. If $e=gsg^{-1}$ for $s,g \in G$, then necessarily $s=e$. So $[e]=\{ e \}$. Next consider $\tau$. Observe that $\tau = \sigma \tau^{2} \sigma$; thus, $\tau \sim \tau^{2}$. Since [[conjugate#^a1150e|the order of elements in a conjugacy class is constant]], no other element can be [[conjugate]] to $\tau$, so $[\tau]=\{ \tau, \tau^{2} \}$. [[Equivalence class]]es [[partition]] sets, so $[\tau^{2}]=\{ \tau,\tau^{2} \}$ too. Finally, we see that $\sigma \tau^{2}= (\tau) \sigma \tau (\tau^{2})$ implying $\sigma \tau \sim \sigma \tau^{2}$ and that $\sigma \tau = \tau \sigma \tau^{2}$, thus $\sigma \tau \sim \sigma$. Thus the final [[equivalence class]] is $[\sigma]=[\sigma \tau]=[\sigma \tau^{2}]=\{ \sigma, \sigma \tau^{2} , \sigma \tau\}$. ^example-1 > [!basicexample] Example. (Conjugacy classes in $D_{4}$) > Consider the [[dihedral group]] $\{ e,\tau,\tau^{2},\tau^{3} , \sigma, \sigma \tau^{2}, \sigma \tau^{3}\}$. > One can verify the [[conjugate|conjugacy]] classes are $\{ e \}, \{ \tau, \tau^{3} \}, \{ \tau^{2} \},\{ \sigma, \sigma \tau^{2} \}, \{ \sigma \tau,\sigma \tau^{2} \}$. These correspond to 'doing nothing', 'one-turns', 'upside-downs', 'horizontal\vertical reflection', 'reflection over corners'. > \ > For more generality, see [[conjugacy classes of the dihedral group]]. ^example-2 ---- #### ---- #### 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 > ```