Conjugacy Classes of a Group

# Conjugacy Classes of a Group

 Proposition 1: Let $G$ be a group. Then conjugacy of elements of $G$ is an equivalence relation on $G$.
• Proof: Let $\sim$ be the relation on $G$ defined for all $g_1, g_2 \in G$ by $g_1 \sim g_2$ if and only if $g_1$ is a conjugate of $g_2$.
• Reflexivity: For each $g \in G$ we have that $g = ege^{-1}$ where $e \in G$ denotes the identity in $G$. So for each $g \in G$, $g \sim g$.
• Symmetry: Suppose that $g_1 \sim g_2$. Then there exists an $a \in G$ such that $g_1 = ag_2a^{-1}$. So $a^{-1}g_1a = g_2$, i.e., $g_2 = (a^{-1})g(a^{-1})^{-1}$ Thus $g_2 \sim g_1$. So for all $g_1, g_2 \in G$ if $g_1 \sim g_2$ then $g_2 \sim g_1$.
• Transitivity: Suppose that $g_1 \sim g_2$ and $g_2 \sim g_3$. Then there exists an $a \in G$ such that $g_1 = ag_2a^{-1}$ and there exists a $b \in G$ such that $g_2 = bg_3b^{-1}$. So:
(1)
\begin{align} \quad g_1 = ag_2a^{-1} = a(bg_3b^{-1})a^{-1} = (ab)g_3(ab)^{-1} \end{align}
• So $g_1 \sim g_3$. So for all $g_1, g_2, g_3 \in G$ if $g_1 \sim g_2$ and $g_2 \sim g_3$ then $g_1 \sim g_3$.
• Therefore $\sim$ is an equivalence relation on $G$. $\blacksquare$

Recall that an equivalence relation on a set partitions that set into groups called equivalence classes. From Proposition 1 above, the equivalence relation of conjugacy partitions the group $G$ into equivalence classes, and we give these equivalence classes a special name.

 Definition: Let $G$ be a group. The Conjugacy Class of $g \in G$ is the equivalence class of $g$ corresponding to the equivalence relation of conjugacy on $G$, i.e., the conjugacy class of $g \in G$ is the set $\{ h \in G : h = aga^{-1} \: \mathrm{for \: some \:} a \in G \}$. A Trivial Conjugacy Class is a conjugacy class containing only one element.