# Conjugacy in a Group

Conjugate Element: If $a,b \in G$, then $b$ is said to be a conjugate of $a$ in $G$ if there exists an element $x \in G$ such that $b = {x^{ – 1}}ax$.

Symbolically, we shall write $a \sim b$ for this and shall refer to this relation as conjugacy.

Then $b \sim a \Leftrightarrow b = {x^{ – 1}}ax$ for some $x \in G$

Theorem: Conjugacy is an equivalence relation in a group.

Proof:

(i) Reflexivity: Let $a \in G$, then $a = {e^{ – 1}}ae$, hence $a \sim a\,\,\,\forall a \in G$, i.e. the relation of conjugacy is reflexive.

(ii) Symmetric: Let $a \sim b$ so that there exists an element $x \in G$ such that $a = {x^{ – 1}}bx,\,\,\,\,a,b \in G$. Now
$\begin{gathered} a \sim b \Rightarrow a = {x^{ – 1}}bx \\ \,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow xa = x\left( {{x^{ – 1}}bx} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow xa{x^{ – 1}} = \left( {x{x^{ – 1}}} \right)b\left( {x{x^{ – 1}}} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow b = xa{x^{ – 1}} \\ \,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow b = {\left( {{x^{ – 1}}} \right)^{ – 1}}a{x^{ – 1}},\,\,\,\,x \in G \\ \,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow b \sim a \\ \end{gathered}$

Thus $a \sim b = b \sim a$. Hence the relation is symmetric.

(iii) Transitivity: Let there exist two elements $x,y \in G$ such that $a = {x^{ – 1}}bx$ and $b = {y^{ – 1}}cy$ for $a,b,c \in G$. Hence $a \sim b$, $b \sim c$
$\begin{gathered} \Rightarrow a = {x^{ – 1}}bx\,\,{\text{and}}\,\, \Rightarrow b = {x^{ – 1}}cx \\ \Rightarrow a = {x^{ – 1}}\left( {{y^{ – 1}}cy} \right)x \\ \Rightarrow a = \left( {{x^{ – 1}}{y^{ – 1}}} \right)c\left( {yx} \right) \\ \Rightarrow a = {\left( {yx} \right)^{ – 1}}c\left( {yx} \right) \\ \end{gathered}$

Here $yx \in G$ and $G$ are the group. Therefore$a \sim b,\,\,b \sim c\,\, \Rightarrow a \sim c$.

Hence the relation is transitive.

Thus conjugacy is an equivalence relation on $G$.

Conjugate Classes: For $a \in G$, let $C\left( a \right) = \left\{ {x:x \in G\,\,{\text{and}}\,\,a \sim x} \right\}$, $C\left( a \right)$, the equivalence class of $a$ in $G$ under a conjugacy relation is usually called the conjugate class of $a$ in $G$. It consists of the set of all distinct elements of the type ${y^{ – 1}}ay$.