# 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 exist 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 exist an element $x \in G$ such that $a = {x^{ - 1}}bx,\,\,\,\,a,b \in G$. Now

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 $a \sim b$, $b \sim c$

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

Hence relation is transitive.

Thus conjugacy is 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 conjugacy relation is usually called the conjugate class of $a$ in $G$. It consists of the set of all distinct elements of the type ${y^{ - 1}}ay$.