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. Thereforea \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.