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.


(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 $${y^{ – 1}}ay$$.